Burgers Turbulence
BurgersTurbulence
Jérémie Bec
Laboratoire Cassiopée UMR6202, CNRS, OCA; BP4229, 06304 Nice Cedex 4, France.
Konstantin Khanin
Department of Mathematics, University of Toronto, M5S 3G3 Toronto, Ontario, Canada.
Abstract
The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original
one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more precisely on the
problem of Burgers turbulence, that is the study of the solutions to the one- or multi-dimensional Burgers equation with random
initial conditions or random forcing. Such work was frequently motivated by new emerging applications of Burgers model to
statistical physics, cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the simplest instances of a
nonlinear system out of equilibrium. The study of random Lagrangian systems, of stochastic partial differential equations and
their invariant measures, the theory of dynamical systems, the applications of field theory to the understanding of dissipative
anomalies and of multiscaling in hydrodynamic turbulence have benefited significantly from progress in Burgers turbulence.
The aim of this review is to give a unified view of selected work stemming from these rather diverse disciplines.
Key words: Burgers equation, turbulence, Lagrangian systems.
Contents
1 From interface dynamics to cosmology 2
1.1 The Burgers equation in statistical mechanics 2
1.2 The adhesion model in cosmology 3
1.3 A benchmark for hydrodynamical turbulence 3
2 Basic tools 4
2.1 Inviscid limit and variational principle 4
2.2 Variational principle for the viscous case 5
2.3 Singularities of Burgers turbulence 5
2.4 Remarks on numerical methods 7
3 Decaying Burgers turbulence 8
3.1 Geometrical constructions of the solution 8
Email addresses: jeremie.bec@obs-nice.fr (Jérémie
Bec), khanin@math.toronto.edu (Konstantin Khanin).
3.2 Kida’s law for energy decay 10
3.3 Brownian initial velocities 12
4 Transport of mass in the Burgers/adhesion
model 14
4.1 Mass density and singularities 15
4.2 Evolution of matter inside shocks 16
4.3 Connections with convex optimization
problems 19
5 Forced Burgers turbulence 21
5.1 Stationary régime and global minimizer 21
5.2 Topological shocks 22
5.3 Hyperbolicity of the global minimizer 24
5.4 The case of extended systems 26
6 Time-periodic forcing 29
Preprint submitted to Physics Reports 1 February 2008
http://arXiv.org/abs/0704.1611v1
6.1 Kicked Burgers turbulence 29
6.2 Connections with Aubry–Mather theory 33
7 Velocity statistics in randomly forced
Burgers turbulence 37
7.1 Shocks and bifractality – a replica variational
approach 38
7.2 Dissipative anomaly and operator product
expansion 39
7.3 Tails of the velocity gradient PDF 41
7.4 Self-similar forcing and multiscaling 43
8 Concluding remarks and open questions 45
Acknowledgements 46
References 46
1 From interface dynamics to cosmology
At the end of the thirties, the Dutch scientist J.M. Burg-
ers [26] introduced a one-dimensional model for pressure-
less gas dynamics. He was hoping that the use of a simple
model having much in common with the Navier–Stokes
equation would significantly contribute to the study of
fluid turbulence. This model now known as the Burgers
equation
∂tv + v ∂xv = ν ∂
xv, (1.1)
has not only the same type of hydrodynamical (or advec-
tive) quadratic nonlinearity as the Navier–Stokes equa-
tion that is balanced by a diffusive term, but it also has
similar invariances and conservation laws (invariance un-
der translations in space and time, parity invariance,
conservation of energy and momentum in one dimension
for ν = 0).
Such hopes appeared to be shattered when in the fifties,
Hopf [67] and Cole [33] showed that the Burgers equation
can be integrated explicitly. This model thus lacks one
of the essential properties of Navier–Stokes turbulence:
sensitivity to small perturbations in the initial data and
thus the spontaneous arise of randomness by chaotic dy-
namics. Unable to cope with such a fundamental aspect,
the Burgers equation then lost its interest in “explain-
ing” fluid turbulence.
In spite of this, the Burgers equation reappeared in the
eighties as the asymptotic form of various nonlinear dis-
sipative systems. Physicists and astrophysicists then de-
voted important effort to the understanding of its multi-
dimensional form and to the study of its random solu-
tions arising from random initial conditions or a random
forcing. The goal of this paper is to review selected works
that exemplify this strong renewal of interest in Burgers
turbulence.
The rest of this section is dedicated to the description of
several physical situations where the Burgers equation
arises. We will then see in section 2 that in any dimen-
sion and in the limit of vanishing viscosity, the solutions
to the Burgers equation can be expressed in an explicit
manner in the decaying case or in an implicit manner in
the forced case, in terms of a variational principle that
permits a systematic classification of its various singu-
larities (shocks and others) and of their local structure
(normal form). Section 3 is dedicated to the study of the
decay of the solutions to the one-dimensional unforced
Burgers equation with random initial data. The multi-
dimensional decaying problem is discussed in section 4.
The motivation comes from cosmology where large-scale
structures can be described in terms of mass transport
by solutions to the Burgers equation. The basic prin-
ciples of the forced Burgers turbulence are discussed
in section 5 where the notions of global minimizer and
topological shocks are introduced. Section 6 is dedicated
to the study of the solutions to the periodically kicked
Burgers equation and their relation with Aubry–Mather
theory for commensurate-incommensurate phase transi-
tions. Section 7 reviews various studies of the stochas-
tically forced Burgers equation in one dimension with a
particular emphasize on questions that are arising from
the statistical study of turbulent flows. Finally, section 8
encompasses concluding remarks and a non-exhaustive
list of open questions on the problem of the Burgers tur-
bulence.
1.1 The Burgers equation in statistical mechanics
The Burgers equation appears in condensed matter, in
statistical physics, and also beyond physics in vehicle
traffic models (see [32], for a review on this topic). When
a random forcing term is added - usually a white noise
in time - it is used to describe various problems of in-
terface deposition and growth (see, for instance, [5]). An
instance frequently studied is the Kardar–Parisi–Zhang
(KPZ) model [74]. This continuous version of ballistic
deposition models accounts for the lateral growth of the
interface. Let us indeed consider an interface where par-
ticles deposit with a random flux F that depends both
on time t and on the horizontal position ~x. The growth
of the local height h happens in the direction normal to
the interface and its time evolution is given by
|∇h|2 = ν∇2h+ F, (1.2)
where the first term of the right-hand side represents the
relaxation due to a surface tension ν. The gradient of
(1.2) gives the multidimensional Burgers equation
∂t~v + ~v · ∇~v = ν∇2~v −∇F, ~v = −∇h, (1.3)
forced by the random potential F . As we will see later,
shocks generically appear in the solution to the Burgers
equation in the inviscid limit ν → 0. They correspond
to discontinuities of the derivative of the height h. The
KPZ model is hence frequently used to understand the
appearence of roughness in various interface problems, as
for instance front propagation in randomly distributed
forests (see, e.g., [101]).
The Hopf–Cole transformation Z = exp(h/2ν) allows
rewriting (1.2) as a linear problem with random coeffi-
cients.
∂tZ = ν∇2Z +
F Z, (1.4)
This equation appears in many complex systems, as for
instance directed polymers in random media [75,22].
Indeed the solution Z(~x, t) is exactly the partition
function of an elastic string in the random potential
(1/2ν)F (~x, t), subject to the constraint that its bound-
ary is fixed at (~x, t). Note that here, the time variable t
is actually a space variable in the main direction of the
polymer.
1.2 The adhesion model in cosmology
The multidimensional Burgers equation has important
applications in cosmology where it is closely linked to
what is usually referred to as the Zel’dovich approxi-
mation [112]. In the limit of vanishing viscosity ν → 0
the Burgers equation is known as the adhesion model
[62]. Right after the decoupling between baryons and
photons, the primitive Universe is a rarefied medium
without pressure composed mainly of non-collisional
dust interacting through Newtonian gravity. The initial
density of this dark matter fluctuates around a mean
value ρ̄. These fluctuations are responsible for the for-
mation of the large-scale structures in which both the
dark non-baryonic matter and the luminous baryonic
matter concentrate. A hydrodynamical formulation of
the cosmological problem leads to a description where
matter evolves with a velocity ~v, solution of the Euler–
Poisson equation (see, e.g., [98], for further details).
In the linear theory of the gravitational instability, that
is for infinitesimally small initial fluctuations of the den-
sity field, an instability is obtained with potential dom-
inant modes (i.e. ~v = −∇Ψ) and, in the suitable coordi-
nates, the gravitational interactions can be neglected. In
1970, Zel’dovich proposed to extend these two proper-
ties to the nonlinear régimes where density fluctuations
become important. In this approximation, he also pos-
tulates that the acceleration is a Lagrangian invariant,
leading to the formation of caustics. N-body simulations
however show that the large-scale structures of the Uni-
verse are much simpler than caustics: they resemble sort
of thin layers in which the particles tend to be trapped
(see figure 1(a)).
Fig. 1. (a) Projection of the matter distribution in a slice
obtained from an N-body simulation by the Virgo consor-
tium [71]. (b) Composite picture showing the superposition
of the results of an N-body simulation with the skeleton of
the results obtained from the adhesion model (from [78]).
It was shown by Gurbatov and Saichev [62] that these
structures are very well approximated by those obtained
when constraining the particles not to cross each other
but to stick together. Even if this mechanism is not col-
lisional but rather gravitational (probably due to insta-
bilities at small spatial scales), its effect can be modeled
by a small viscous diffusive term in the Euler–Poisson
equation and thus amounts to considering the Burgers
equation in the limit of vanishing viscosity.
1.3 A benchmark for hydrodynamical turbulence
As a nonlinear conservation law, and since its solution
can be easily known explicitly, the one-dimensional
Burgers equation frequently serves as a testing ground
for numerical schemes, and especially for those dedi-
cated to compressible hydrodynamics. For instance, it
is a central example for the validation of finite-volumes
schemes.
The Burgers equation was also used for testing statisti-
cal theories of turbulence. For instance, field theoretical
methods have frequently been applied to turbulence (see
[96,102]). These approaches had very little impact until
recently when they led to significant advances in the un-
derstanding of intermittency in passive scalar advection
(see, e.g., [46] for a review). In the past such attempts
were mostly based on a formal expansion of the non-
linearity using, for instance, Feynman graphs. Since the
Burgers equation has the same type of quadratic nonlin-
earity as the Navier–Stokes equation, such methods are
applicable in both instances. From this point of view,
it is important to know answers for Burgers turbulence
to questions that are generally asked for Navier–Stokes
turbulence. For instance, Burgers turbulence with a ran-
dom forcing is the counterpart of the hydrodynamical
turbulence model where a steady state is maintained by
an external forcing. The Burgers equation has frequently
been used as a model where the dissipation of kinetic
energy remains finite in the limit of vanishing viscosity
(dissipative anomaly). This allows singling out artifacts
arising from manipulation that ignore shock waves (see,
for instance, [51,40]).
Beyond statistical theory, Burgers turbulence gives a
simple hydrodynamical training ground for developing
mathematical tools to study not only Navier–Stokes tur-
bulence but also various hydrodynamical or Lagrangian
problems. The forced Burgers equation has recently been
at the center of studies that allowed unifying different
branches of mathematics. Mainly used in the past as a
simple illustration of the notion of entropy (or viscos-
ity) solution for conservation laws [83,95,85], the Burg-
ers equation was related in the eighties to the theory
of Hamiltonian systems developed by Kolmogorov [80],
Arnold [2] and Moser [93] (KAM), through the introduc-
tion of the weak KAM theory [43,47,48]. More recently,
the study of the solutions to the Burgers equation with a
random forcing was at the center of a “random” Aubry–
Mather theory related to random Lagrangian systems
[38,69]. A particular emphasis on these aspects of Burg-
ers turbulence is given throughout the present review.
For the application of the Burgers equation to the prop-
agation of random nonlinear waves in nondispersive me-
dia, we refer the reader to the book written by Gurba-
tov, Malakhov, and Saichev [61]. For a complete state of
the art on most mathematical apsects of Burgers turbu-
lence, we refer the reader to the lecture notes by Woy-
czyński [110].
2 Basic tools
In this section we introduce various analytical, geomet-
rical and numerical tools that are useful for constructing
solutions to the Burgers equation, with and without forc-
ing, in the limit of vanishing viscosity. All these tools are
derived from a variational principle that allows writing
in an implicit way the solution at any time. This varia-
tional principle leads to a straightforward classification
of the various singularities that are generically present
in the solution to the Burgers equation.
2.1 Inviscid limit and variational principle
We consider here the multidimensional viscous Burgers
equation with forcing
∂t~v + (~v · ∇)~v = ν∇2~v −∇F (~x, t), (2.1)
where ~x lives on a prescribed configuration space Ω of
dimension d. For a potential initial condition, ~v(~x, t0) =
−∇Ψ0(~x), the velocity field remains potential by con-
struction at any later time, ~v = −∇Ψ, where the poten-
tial Ψ satisfies the equation
∂tΨ −
|∇Ψ|2 = ν∇2Ψ + F. (2.2)
Note that if one sets abruptly ν = 0 in (2.2), then −Ψ
solves the Hamilton–Jacobi equation associated to the
Hamiltonian H(~q, ~p) = |~p|2 + F (~q, t). In the unforced
case, −Ψ is a solution of the Hamilton–Jacobi equation
associated to the dynamics of free particles. The Hopf–
Cole transformation [67,33] uses a change of unknown
Ψ(~x, t) = 2ν ln Θ(~x, t). The new unknown scalar field Θ
is solution of the (imaginary-time) Schrödinger equation
∂tΘ = ν∇2Θ +
F Θ, (2.3)
with the initial condition Θ(~x, t0) = exp(Ψ0(~x)/(2ν)).
The solution can be expressed through the Feynman-
Kac formula
Θ(~x, t)=
Ψ0( ~Wt0)−
F ( ~Ws, s) ds
, (2.4)
where the brackets 〈·〉 denote the ensemble average with
respect to the realizations of the d-dimensional Brown-
ian motion ~Ws with variance 2ν defined on the configu-
ration space Ω and which starts at ~x at time t. The limit
ν → 0 is obtained by a classical saddle-point argument.
The main contribution will come from the trajectories
~W minimizing the argument of the exponential; the ve-
locity potential can then be expressed as a solution of
the variational principle
Ψ(~x, t) = − inf
~γ(·)
[A(~γ, t0, t) − Ψ0(~γ(t0))] , (2.5)
where the infimum is taken over all trajectories ~γ that
are absolutely continuous (e.g. piece-wise differentiable)
with respect to the time variable and that satisfy ~γ(t) =
~x. The action A(~γ, t0, t) associated to the trajectory ~γ
is defined by
A(~γ, t0, t) =
|~̇γ(s)|2 − F (~γ(s), s)
ds, (2.6)
where the dot stands for time derivative. The kinetic
energy term |~̇γ|2/2 comes from the propagator of the d-
dimensional Brownian motion ~W . This variational for-
mulation of the solution to the Burgers equation was
obtained first by Hopf [67], Lax [83] and Oleinik [95]
for scalar conservation laws. Its generalization to mul-
tidimensional Hamilton–Jacobi equations was done by
Kruzhkov [82] (see also [85]). In the case of a random
forcing potential F , it was shown by E, Khanin, Mazel
and Sinai [38] that this formulation is still valid after re-
placing the action by a stochastic integral. It is also im-
portant to notice that the variational formulation (2.5)
in the limit of vanishing viscosity is valid irrespective of
the configuration space Ω on which the solution is de-
fined.
The minimizing trajectories ~γ(·) necessarily satisfy at
times s < t the Newton (or Euler–Lagrange) equation
~̈γ = −∇F (~γ(s), s), (2.7)
with the boundary conditions (at the final time t)
~γ(t) = ~x and ~̇γ(t) = ~v(~x, t). (2.8)
Note that these equations are only valid backward in
time. Extending them to times larger than t requires
knowing that the Lagrangian particle will neither cross
the trajectory of another particle, nor be absorbed by
a mature shock. This requires global knowledge of the
solution that satisfies the variational principle (2.5).
When the forcing term is absent from (2.1), it is easily
checked that the variational principle reduces to
Ψ(~x, t) = max
Ψ0(~x0) −
|~x− ~x0|2
, (2.9)
where the maximum is taken over all initial positions
~x0 in the configuration space Ω. The Euler–Lagrange
equation takes then the particularly simple form
~̈γ = 0, i.e. ~x = ~x0 + t ~v0(~x0), (2.10)
which simply means that the initial velocity is conserved
along characteristics.
Typically there exist Eulerian locations ~xwhere the min-
imum in (2.5) – or the maximum in (2.9) in the unforced
case – is reached for several different trajectories ~γ. Such
locations correspond to singularities in the solution to
the Burgers equation. After their appearance, the veloc-
ity potential Ψ contains angular points corresponding to
discontinuities of the velocity field ~v = −∇Ψ.
2.2 Variational principle for the viscous case
The derivation of the variational principle (2.5) makes
use of the Hopf–Cole transformation and of the
Feynman–Kac formula. There is in fact another ap-
proach which also yields a variational formulation of
the solution to the viscous Hamilton–Jacobi equation
(2.2). Indeed it turns out that the solution to (2.2) can
be obtained in the following way. Consider solutions to
the stochastic differential equation
d~γ~u = ~u(~γ~u, s) ds+
2ν d ~Ws , (2.11)
where ~u is a stochastic control, that is an arbitrary time-
dependent velocity field which depends (progressively
measurably) on the noise ~W . Limiting ourselves to so-
lutions satisfying the final condition ~γ~u(t) = ~x, we can
write
Ψ(~x, t) = − inf
〈A~u(~γ~u, t0, t) − Ψ0(~γ~u(t0))〉 , (2.12)
where the brackets 〈·〉 now denote average with respect
to ~Ws and the action is given by
A~u(~γ~u, t0, t) =
|~u(s)|2 − F (~γ~u(s), s)
ds. (2.13)
It is obvious that this variational principle gives (2.5) in
the inviscid limit ν → 0. Note that this approach has the
advantage to be applicable not only to Burgers dynamics
but to any convex Lagrangian (see [50,58]).
2.3 Singularities of Burgers turbulence
The singularities appearing in the course of time play
an essential role in understanding various aspects of the
statistical properties in the inviscid limit. The shocks –
discontinuities of the velocity field – and other singular-
ities, such as preshocks, generally not associated to dis-
continuities, are often responsible for non-trivial univer-
sal behaviors. In order to understand the contribution
of each kind of singularities, it is first important to know
in a detailed manner their genericity and their type.
As we have seen in the previous section, the potential
solutions to the multidimensional Burgers equation can
be expressed in the inviscid limit in terms of the vari-
ational principle (2.5) (that reduces to (2.9) in the un-
forced case). There typically exist Eulerian locations ~x
where the minimum is either degenerate or attained for
several trajectories. A co-dimension can be associated
to such points by counting the number of relations that
are necessary to determine them. The singular locations
of co-dimension c form manifolds of the Eulerian space-
time with dimension (d− c). The singularities with the
lower co-dimension are the shocks corresponding to the
Eulerian positions where two different trajectories min-
imize (2.5); they form Eulerian manifolds of dimension
(d− 1): in one dimension the shocks are isolated points,
in two dimensions they are lines, in three dimensions
surfaces, etc. There also exist Eulerian manifolds with
three different minimizing trajectories. In one dimen-
sion, they are isolated space-time events corresponding
to the merger of two shocks. In two dimensions, they are
triple points where three shock lines meet. In three di-
mensions they are filaments corresponding to the inter-
section of three shock surfaces. There also exist Eulerian
locations where the minimum in (2.5) is reached for four
different trajectories, etc.
points
termination
shock lines
points
triple
Fig. 2. Typical aspect of the singularities present at a fixed
time in the solution for (a) d = 2 and (b) d = 3.
The generic form of such singularities and their typi-
cal metamorphoses occurring in the course of time were
studied in details and classified for d = 2 and d = 3 by
Arnold, Baryshnikov and Bogaevsky in the Appendix
of [62] and in a more detailed paper by Bogaevsky [17].
This classification is based on two criteria: (i) the num-
ber of trajectories minimizing (2.5) and (ii) the multi-
plicity of each of these minima. The shocks correspond-
ing to locations with two distinct minimizers are hence
denoted by A21. At a fixed time, the A
1 singularities are
discrete points in one dimension. In two dimensions (see
figure 2(a)) they form curve segments with extremities
that can be either triple points A31 or isolated termina-
tion points of the type A3 corresponding to a degenerate
minimum. In three dimensions (see figure 2(b)) the sin-
gular manifold is formed by shock surfaces of A21 points.
The boundaries of these surfaces are either made of de-
generate A3 points or of triple lines made of A
1 points.
The triple lines intersect at isolated A41 points or inter-
sect shock boundaries at particular singularities called
A1A3 where the minimum is attained in two points, one
of which is degenerate.
It is important to remark here that degenerate singu-
larities (of the type A3 or of higher orders A5, A7, etc.)
introduce in the solution points where the velocity gra-
dients becomes arbitrarily large. This is not the case of
the An1 singularities which correspond to discontinuities
of the velocity but are associated to bounded values of
its gradients. As we will see in sections 4 and 7, these
degenerate singularities are responsible for an algebraic
behavior of the probability density function of velocity
gradients, velocity increments and of the mass density.
Fig. 3. Illustration of the similarities between the singular
manifold in space time for d = 1 and at fixed time for d = 2
(b). The two manifolds contain the same type of singularities
with the same co-dimensions. The restrictions on the possible
metamorphoses in dimension d = 1 are the following: a point
of the type A3 can only exist at the bottom extremity of
a shock trajectory; the A31 points necessarily correspond to
the merger of two shocks; shock trajectories cannot have a
horizontal tangent.
The singularities with co-dimensions (d+ 1) generically
appear in the solution at isolated times. They corre-
spond to instantaneous changes in the topological struc-
ture of the singular manifold, called metamorphoses and
can be also classified (see [17]). In one dimension, there
are two generic metamorphoses: shock formations (the
preshocks) corresponding to a specific space-time loca-
tion where the minimum is degenerate (A3 singularities)
and shock mergers associated to space-time positions
where the minimum is attained for three different tra-
jectories (A31 singularities). We see that some of the sin-
gularities generically present in two dimensions appear
at isolated times in three dimensions. Actually, all the
singularities generically present in dimension (d+1) ap-
pear in dimension d on a discrete set of space time, that
is at isolated positions and instants of time. However, it
has been shown in [17] that the irreversible dynamics of
the Burgers equation restricts the set of possible meta-
morphoses. The admissible metamorphoses are charac-
terized by the following property: after the bifurcation,
the singular manifold must remain locally contractible
(homotopic to a point in the neighborhood of the Eu-
lerian location of the metamorphosis). This topological
restriction is illustrated for the one-dimensional case in
figure 3. Note that this constraint actually holds for all
solutions to the Hamilton–Jacobi equation in the limit
of vanishing viscosity, as long as the Hamiltonian is a
convex function.
In order to determine precisely how all these singularities
contribute to the statistical properties of the solution, it
is important to know the local structure of the velocity
(or potential) field in their vicinity. Variousnormal forms
can be obtained from the multiplicity of the minimum
in the variational formulation of the solution (2.5). In
the case without forcing, they can be obtained from a
Taylor expansion of the initial velocity potential. This
will be used in next section to determine the tail of the
probability distribution of a mass density field advected
by a velocity solution to the Burgers equation.
2.4 Remarks on numerical methods
All the traditional methods used to solve equations of
fluid dynamics, or more generally any partial differen-
tial equations, can be used to obtain the solutions to the
Burgers equation. However, as we have seen above, the
solution typically has singularities (discontinuities of the
velocity) in the limit of vanishing viscosity. Hence meth-
ods which rely on the smoothness of the solution require
a non-vanishing viscosity, which is introduced either in
an explicit way to ensure stability (as, e.g., for pseudo-
spectral methods) or in an implicit way through the
discretization procedure (as for finite-differences meth-
ods). In both cases the value of the viscosity is deter-
mined from the mesh size and, even in one dimension,
their uses might be very disadvantageous. We will now
demonstrate various numerical methods that allow ap-
proximating the solutions to the Burgers equation di-
rectly in the limit of vanishing viscosity ν → 0.
2.4.1 Finite volumes
The one-dimensional Burgers equation with no forcing is
a scalar conservation law. Its entropic solutions (or vis-
cosity solutions) can thus be approximated numerically
by finite-volume methods. Instead of constructing a dis-
crete approximation of the solution on a grid, such meth-
ods consist in considering an approximation of its mean
value on a discrete partitioning of the system into finite
volumes. One then needs to evaluate for each of these
volumes the fluxes exchanged with each of its neighbors.
Various approximations of these fluxes were introduced
by Godunov, Roe, and Lax and Wendroff (see, e.g., [35],
Vol. 3, for a review). These methods require to dicretize
both space and time. The time step being then related
to the spatial mesh size by a Courant–Friedrichs–Lewy
type condition. Thus to integrate the equation during
times comparable to one eddy turnover time, they re-
quire a computational time O(N2) where N is the res-
olution. As we now show there actually exist numerical
schemes that allow constructing the solution to the de-
caying Burgers equation for arbitrary times without any
need to compute the solution at intermediate times.
2.4.2 Fast Legendre transform
As we have seen in section 2.1, the solution to the un-
forced Burgers equation is given explicitly by the varia-
tional principle (2.9). A method based on the idea of us-
ing this formulation together with a monotonicity prop-
erty of the Lagrangian map ~x0 → ~x = ~X(~x0, t) was
given in [94]. It is called the fast Legendre transform
whose principles were already sketched in [23]. Both Eu-
lerian and Lagrangian positions are discretized on reg-
ular grids. Then, for a fixed Eulerian location ~x(i) on
the grid, one has to find the corresponding Lagrangian
coordinate ~x
0 maximizing (2.9). A naive implementa-
tion would require O(NdEN
L) operations if the Eulerian
and the Lagrangian grids contain NdE and N
L points re-
spectively. Actually the number of operations can be re-
duced to O((NE lnNL)
d) by using the monotonicity of
the Lagrangian map, that is the fact that for any pair
of Lagrangian positions ~x
0 and ~x
0 , one has at any
time [ ~X(~x
0 , t) − ~X(~x
0 , t)] · (~x
0 − ~x
0 ) ≥ 0. In the
case of orthogonal grids, this property allows perform-
ing the maximization by exploring along a binary tree
the various possibilities; thus the number of operations
is reduced to lnNL for each of the NE positions on the
Eulerian grid. Such algorithms give access to the solu-
tion not only directly in the limit of vanishing viscosity
but also by jumping directly from the initial time to an
arbitrary time.
This method can also be used for the forced Burgers
equation, approximating the forcing by a sum of im-
pulses at discrete times and letting the solution decay
between two such kicks. This gives an efficient algorithm
for the forced Burgers equation directly applicable in the
limit of vanishing viscosity.
2.4.3 Particle tracking methods
In one dimension, Lagrangian methods can be imple-
mented in a straightforward manner after noticing that
particles cannot cross each other and that it is advis-
able to track not only fluid particles but also shocks (see,
e.g., [6]). Lagrangian methods can in principle be used
to solve the Burgers equation in any dimension. How-
ever the shock dynamics is meaningful only for poten-
tial solutions. Outside the potential framework almost
nothing is known about the construction of the solution
beyond the first crossing of trajectories. In the potential
case, a particle method can be formulated by choosing
to represent the solution in the position-potential (~x,Ψ)
space instead of the position-velocity (~x,~v) space. An
idea in two dimensions, which was not yet implemented,
consists in considering a meshing of the hyper-surface
defined by the velocity potential. If such a mesh contains
only triple points, such singularities are preserved by the
dynamics and can be tracked using the results discussed
below in section 4.2 and by checking at all time steps in
an exhaustive manner at all the metamorphoses encoun-
tered by triple points.
3 Decaying Burgers turbulence
We focus in this section on the solutions to the d-
dimensional unforced potential Burgers equation
∂t~v + ~v ·∇~v = ν∇2~v, ~v(~x, 0)=~v0(~x)=−∇Ψ0(~x). (3.1)
As showed in section 2.1, the solution can be expressed
in the limit of vanishing viscosity ν → 0 in terms of a
variational principle that relates the velocity potential
at time t to its initial value:
Ψ(~x, t) = max
Ψ0(~x0) −
|~x− ~x0|2
. (3.2)
The next subsection describes several geometrical con-
structions of the solution that are helpful to determine
various statistical properties of the decaying prob-
lem (3.1). This is illustrated in subsections 3.2 and 3.3
which are devoted to the study of the decay of smooth
homogeneous and of Brownian initial data, respectively.
The study of the solutions to the Burgers equation trans-
porting a density field is of particular interest in the ap-
plication of the Burgers equation in cosmology within
the framework of the adhesion model. This question will
be discussed in section 4.
3.1 Geometrical constructions of the solution
3.1.1 The potential Lagrangian manifold
The variational formulation of the solution (3.2) has
a simple geometrical interpretation in the position-
potential space (~x,Ψ) of dimension d + 1. Indeed, con-
sider the d-dimensional manifold parameterized by the
Lagrangian coordinate ~x0 and defined by
~x = ~x0 − t∇Ψ0(~x0)
Ψ = Ψ0(~x0) −
|∇Ψ0(~x0)|2.
(3.3)
The first line corresponds to the position where the gra-
dient of the argument of the maximum function in (3.2)
vanishes while the second line is just its argument eval-
uated at the maximum. For a sufficiently regular initial
potential Ψ0 (at least twice differentiable) and for suf-
ficiently small times, equation (3.3) unambiguously de-
fines a single-valued function Ψ(~x, t). However, there ex-
ists generically a time t⋆ at which the manifold is folding.
Figure 4(a) (upper) shows in one space dimension the
typical shape of the Lagrangianmanifold defined by (3.3)
after the critical time t⋆. For some Eulerian positions ~x,
there is more than one branch and cusps are present at
Eulerian locations where the number of branches change.
The situation is very similar in higher dimensions as il-
lustrated for d = 2 in figure 4(b). Clearly from the varia-
tional principle (2.9), the correct solution to the inviscid
Burgers equation is obtained by taking the maximum,
that is the highest branch. The velocity potential is by
construction always continuous but it contains angular
points corresponding to discontinuities of the velocity
~v = −∇Ψ. Such singularities are located at Eulerian lo-
cations where the maximum in (2.9) is degenerate and
attained for different ~x0. As already discussed in sec-
tion 2.3 the different singularities appearing in the solu-
tions can be classified in any dimension.
Below we describe other geometrical constructions of the
solutions to the decaying Burgers equation in the limit
of vanishing viscosity that are based on the variational
principle (2.9).
3.1.2 The velocity Lagrangian manifold
In one dimension, when the velocity field is always po-
tential, the method based on the study of the poten-
tial manifold in the (x,Ψ) space described above can
be straightforwardly extended to the position-velocity
phase space. Consider the Lagrangian manifold defined
x = x0 − t v0(x0)
v = v0(x0).
(3.4)
Maxwell
   rule
Fig. 4. (a) Lagrangian manifold for d = 1 in the (x,Ψ)
plane (upper) and in the (x, v) plane (lower); the heavy lines
correspond to the correct Eulerian solutions. (b) Lagrangian
manifold in the (~x,Ψ) space for d = 2.
The regular parts of the graph of the solution are nec-
essarily contained in this manifold. However, for times
larger than t⋆, folding appears and the naive solution
would be multi-valued. To construct the true solution
one should find a way to choose among the different
branches. In one dimension, there is a simple relation
between the potential Lagrangian manifold in the (x,Ψ)
plane and those of the (x, v) plane defined by (3.4):
the potential manifold is obtained by taking the “multi-
valued integral” that can be defined by transforming the
spatial integral into an integral with respect to the arc
length. The maximum representation (2.9) implies that
the velocity potential is continuous. Hence a shock cor-
responds to an Eulerian position x where two points be-
longing to different branches define equal areas in the
(x, v) plane. In the case of a single loop of the manifold,
this is equivalent to applying the Maxwell rule to deter-
mine the shock position (see figure 4(a) - lower). This
construction of the solution can become rather involved
as soon as the number of shocks becomes large or that
several mergers have taken place. For the moment there
xshock
interval
regular
points
Φ, Φc
Fig. 5. Convex hull construction in terms of the Lagrangian
potential (a) for d = 1 and (b) for d = 2.
is no generalization to dimension higher than one of this
Maxwell rule construction of the solution. For such an
extension, one needs to develop a geometrical framework
to describe the Lagrangian manifold in the (~x,~v) space.
Such approaches would certainly shed some light on the
problem of constructing non-potential solutions to the
Burgers equation in the limit of vanishing viscosity.
3.1.3 The convex hull of the Lagrangian potential
Another geometrical construction of the solution, which
is valid in any dimension makes use of the Lagrangian
potential
Φ(~x0, t) = tΨ0(~x0) −
|~x0|2
. (3.5)
Clearly, the negative gradient of the Lagrangian poten-
tial gives the naive Lagrangian map
~X(~x0, t) = −∇~x0Φ(~x0, t) = ~x0 + t~v0(~x0), (3.6)
that is satisfied by Lagrangian trajectories as long as
they do not enter shocks. The maximum formulation of
the solution (2.9) can be rewritten as
tΨ(~x, t) +
|~x|2
= max
(Φ(~x0) + ~x0 · ~x), (3.7)
which represents the potential as, basically, a Legendre
transform of the Lagrangian potential. An important
property of the Legendre transform is that the right-
hand side. of (3.7) is unchanged if the Lagrangian po-
tential Φ is replaced by its convex hull, that is the in-
tersection of all the half planes containing its graph. In
other terms, the convex hull Φc of the Lagrangian po-
tential Φ is defined as Φc(~x0, t) = inf g(~x0), where the
infimum is taken over all convex functions g satisfying
g(·) ≥ Φ(·, t). This is illustrated in one dimension in fig-
ure 5(a) which shows both regular points (Lagrangian
points which have not fallen into a shock) and one shock
interval, situated below the segment which is a part of
the convex hull. In two dimensions, as illustrated in fig-
ure 5(b), the convex hull is typically formed by regular
points, by ruled surfaces, and by triangles which corre-
spond, to the regular part of the velocity field, the shock
lines, and the shock nodes, respectively.
Note that in one dimension, there exists an equivalent
construction which is directly based on the Lagrangian
map x0 7→ X(x0, t) defined by (3.6). Working with the
convex hull is equivalent to the Maxwell rule applied
to the non-invertible regions of the Lagrangian map. A
shock corresponds to a whole Lagrangian interval having
a single point as an Eulerian image. One then talks about
a Lagrangian shock interval.
3.1.4 The paraboloid construction
Finally, the maximum representation (3.7) leads in a
straightforwardway to another geometrical construction
of the solution. As illustrated in figure 6 in both one and
two dimensions, a paraboloid with apex at ~x and ra-
dius of curvature proportional to t is moved down in the
(~x0,Ψ0) space until it touches the surface defined by the
initial velocity potential Ψ0 at the Lagrangian location
associated to ~x. The location ~x0 where the paraboloid
touches the graph of the potential is exactly the pre-
image of ~x. If it touches simultaneously at several loca-
tions, a shock is located at the Eulerian position ~x. One
constructs in this way the inverse Lagrangian map.
3.2 Kida’s law for energy decay
An important issue in turbulence is that of the law of
decay at long times when the viscosity is very small.
Before turning to the Burgers equation it is useful to re-
call some of the features of decay for the incompressible
Navier–Stokes case. It is generally believed that high-
Reynolds number turbulence has universal and non-
trivial small-scale properties. In contrast, large scales,
important for practical applications such as transport of
heat or pollutants, are believed to be non-universal. This
is however so only for the toy model of turbulence main-
tained by prescribed large-scale random forces. Very
high-Reynolds number turbulence, decaying away from
its production source, and far from boundaries can relax
Fig. 6. Paraboloid construction of solution for (a) d = 1 and
(b) d = 2.
under its internal nonlinear dynamics to a (self-similarly
evolving) state with universal and non-trivial statistical
properties at all scales. Von Kármán and Howarth [109],
investigating the decay for the case of high-Reynolds
number homogeneous isotropic three-dimensional tur-
bulence, proposed a self-preservation (self-similarity)
ansatz for the spatial correlation function of the ve-
locity: the functional shape of the correlation function
remains fixed, while the integral scale L(t) grows in
time and the mean kinetic energy E(t) = V 2(t) decays,
both following power laws; there are two exponents
which can be related by the condition that the energy
dissipation per unit mass |Ė(t)| should be proportional
to V 3/L. But an additional relation is needed to actu-
ally determine the exponents. The invariance in time of
the energy spectrum at low wavenumbers, known as the
“permanence of large eddies” [53,84,63] can be used to
derive the law of self-similar decay when the initial spec-
trum E0(k) ∝ kn at small wavenumbers k with n below
a critical value equal to 3 or 4, the actual value being
slope 1/t
Random
position
Fig. 7. Snapshot of solution of decaying Burgers turbulence
at long times when spatial periodicity is assumed.
disputed because of the “Gurbatov phenomenon” (see
the end of this section). One then obtains a law of decay
E(t) ∝ t−2(n+1)/(3+n). (Kolmogorov [79] proposed a law
of energy decay V 2(t) ∝ t−10/7, which corresponds to
n = 4 and used in its derivation the so-called “Loitsyan-
sky invariant”, a quantity actually not conserved, as
shown by Proudman and Reid [100].) When the initial
energy spectrum at low wavenumbers goes to zero too
quickly, the permanence of large eddies cannot be used,
because the energy gets backscattered to low wavenum-
bers by nonlinear interactions. For Navier–Stokes tur-
bulence the true law of decay is then known only within
the framework of closure theories (see, e.g., [84]).
For one-dimensional Burgers turbulence, many of the
above issues are completely settled. First, we observe
that the problem of decay is quite simple if spatial peri-
odicity is assumed. Indeed, all the shocks appearing in
the solution will eventually merge into a single shock per
period, as shown in figure 7. The position of this shock
is random and the two ramps have slope 1/t, as is easily
shown using the parabola construction of subsection 3.1.
Hence, the law of decay is simply E(t) ∝ t−2. Nontrivial
laws of decay are obtained if the Burgers turbulence
is homogeneous in an unbounded domain and has the
“mixing” property (which means, roughly, that correla-
tions are vanishing when the separation goes to infinity).
The number of shocks is then typically infinite but their
density per unit length decreases in time because shocks
are constantly merging. The E(t) ∝ t−2(n+1)/(3+n) law
mentioned above can be derived for Burgers turbulence
from the permanence of large eddies when n ≤ 1 [63].
For n = 0, this t−2/3 law was actually derived by Burg-
ers himself [27].
The hardest problem is again when permanence of
large eddies does not determine the outcome, namely
for n > 1. This problem was solved by Kida [77] (see
also [51,61,63]).
We now give some key ideas regarding the derivation of
Kida’s law of energy decay. We assume Gaussian, homo-
geneous smooth initial conditions, such that the poten-
tial is homogeneous. Note that a homogeneous function
is not, in general, the derivative of another homogeneous
function. Here this is guaranteed by assuming that the
Fig. 8. An initial potential which is everywhere below the
parabola x20/(2t) + Ψ. The probability of such events gives
the cumulative probability to have a potential at time t less
than Ψ.
initial spectrum of the kinetic energy is of the form
E0(k) ∝ kn for k → 0 with n > 1 . (3.8)
This condition implies that the mean square initial po-
tential
k−2E0(k) dk has no infrared (small-k) diver-
gence (the absence of an ultraviolet divergence is guar-
anteed by the assumed smoothness).
A very useful property of decaying Burgers turbulence,
which has no counterpart for Navier–Stokes turbulence,
is the relation
E(t) =
〈Ψ〉 , (3.9)
which follows by taking the mean of the Hamilton–
Jacobi equation for the potential (2.2) in the absence of
viscosity and of a driving force. Hence, the law of energy
decay can be obtained from the law for the mean po-
tential. The latter can be derived from the cumulative
probability of the potential which, by homogeneity, does
not depend on the position. By (2.9), its expression at
x = 0 is
Prob{Pot.<Ψ}=Prob
∀x0, Ψ0(x0)<
. (3.10)
Expressed in words, we want to find the probability
that the initial potential does not cross the parabola
x20/(2t) + Ψ (see figure 8). Since, at large times t, the
relevant Ψ is going to be large, the problem becomes
that of not crossing a parabola with small curvature and
very high apex. The crossings, more precisely the up-
crossings, are spatially quite rare and, for large t, form
a Poisson process [92] for which
Prob. no crossing ≃ e−〈N(t)〉, (3.11)
where 〈N(t)〉 is the mean number of up-crossings. By
the Rice formula (a consequence of the identity δ(λx) =
(1/|λ|)δ(x)),
〈N(t)〉=
〈∫ +∞
dx0 δ(m(x0)−Ψ)
, (3.12)
where H is the Heaviside function and
m(x0) ≡ Ψ0(x0) −
. (3.13)
Since Ψ0(x0) is Gaussian, the right-hand side of (3.12)
can be easily expressed in terms of integrals over the
probability densities of Ψ0(x0) and of dΨ0(x0)/dx0 (as
a consequence of homogeneity these variables are uncor-
related and, hence, independent). The resulting integral
can then be expanded by Laplace’s method for large t,
yielding
〈N(t)〉 ∼ t1/2Ψ−1/2e−Ψ
, t→ ∞. (3.14)
When this expression is used in (3.11) and the result is
differentiated with respect to Ψ to obtain the probability
density function (PDF) of p(Ψ), the latter is found to
be concentrated around Ψ⋆ = (ln t)
1/2. It then follows
that, at large times, we obtain Kida’s log-corrected 1/t
law for the energy decay
〈Ψ〉 ∼ (ln t)1/2, E(t) ∼
t(ln t)1/2
, L(t) ∼
. (3.15)
The Eulerian solution, at large times, has the ramp
Fig. 9. The Eulerian solution at large times t. The ramps
have slope 1/t. In time-independent scales, the figure would
be stretched horizontally and squeezed vertically by a factor
proportional to t.
structure shown in figure 9 with shocks of typical
strength V (t) = E1/2(t), separated by a distance L(t).
The fact that Kida’s law is valid for any n > 1, and not
just for n ≥ 2 as thought originally, gives rise to an inter-
esting phenomenon now known as the “Gurbatov effect”:
if 1 < n < 2 the large-time evolution of the energy spec-
trum cannot be globally self-similar. Indeed, the perma-
nence of large eddies, which is valid for any n < 2 dic-
tates that the spectrum should preserve exactly its initial
n behavior at small wavenumbers k, with a constant-
in-time Cn. Global self-similarity would then imply a
t−2(n+1)/(3+n) law for the energy decay, which would
contradict Kida’s law. Actually, as shown in [63], there
are two characteristic wavenumbers with different time
dependences, the integral wavenumber kL(t) ∼ (L(t))−1
and a switching wavenumber ks(t) below which holds
the permanence of large eddies. It was shown that the
same phenomenon is present also in the decay of a pas-
sive scalar [45]. Whether or not a similar phenomenon is
present in three-dimensional Navier–Stokes incompress-
ible turbulence, or even in closure models, is a contro-
versial matter [44,97].
For decaying Burgers turbulence, if we leave aside the
Gurbatov phenomenon which does not affect energy-
carrying scales, the following may be shown. If we rescale
distances by a factor L(t) and velocity amplitudes by a
factor V (t) = E1/2(t) and then let t → ∞, the spatial
(single-time) statistical properties of the whole random
velocity field become time-independent. In other words,
there is a self-similar evolution at large times. Hence,
dimensionless ratios such as the velocity flatness
F (t) ≡
v4(t)
[〈v2(t)〉]2
(3.16)
have a finite limit as t → ∞. A similar property holds
for the decay of passive scalars [28]. We do not know if
this property holds also for Navier–Stokes incompress-
ible turbulence or if, say, the velocity flatness grows with-
out bounds at large times.
3.3 Brownian initial velocities
Initial conditions in the Burgers equation that are Gaus-
sian with a power-law spectrum ∝ k−α have been fre-
quently studied because they belong in cosmology to the
class of scale-free initial conditions (see [98,34]). We con-
sider here the one-dimensional case with Brownian mo-
tion as initial velocity, corresponding to α = 2.
Brownian motion is continuous but not differentiable;
thus, shocks appear after arbitrarily short times and are
actually dense (see figure 10(a)). Numerically supported
conjectures made in [104] have led to a proof by Sinai
[105] of the following result: in Lagrangian coordinates,
the regular points, that is fluid particles which have not
yet fallen into shocks, form a fractal set of Hausdorff
dimension 1/2. This implies that the Lagrangian map
forms a Devil’s staircase of dimension 1/2 (see figure 11).
Note that when the initial velocity is Brownian, the La-
grangian potential has a second space derivative delta-
correlated in space; this can be approximately pictured
as a situation where the Lagrangian potential has very
wild oscillations in curvature. Hence, it is not surprising
that very few points of its graph can belong to its convex
hull (see figure 10(b)).
We will now highlight some aspects of Sinai’s proof of
this result. The idea is to use the construction of the
solution in terms of the convex hull of the Lagrangian
potential (see section 3.1), so that regular points are ex-
actly points where the graph of the Lagrangian poten-
tial coincides with its convex hull. For this problem, the
Hausdorff dimension of the regular points is also equal
to its box-counting dimension, which is easier to deter-
mine. One obtains it by finding the probability that a
−5 −2.5 0 2.5 5
t = 0
t = 1
Fig. 10. Snapshot of the solution resulting from Brownian
initial data in one dimension. (a) Velocity profile at initial
time t = 0 and at time t = 1; notice the dense proliferation
of shocks. (b) Lagrangian potential together with its convex
hull.
Fig. 11. The Lagrangian map looks like a Devil’s staircase:
it is constant almost everywhere, except on a fractal Can-
tor-like set (from [107]).
small Lagrangian interval of length ℓ contains at least
one regular point which belongs simultaneously to the
graph of the Lagrangian potential Φ and to its convex
hull. In other words, one looks for points, such asR, with
the property that the graph of Φ lies below its tangent
at R (see figure 12). Following Sinai, this can be equiva-
lently formulated by the box construction with the fol-
lowing constraints on the graph:
Left: graph of the potential below the half line Γ−,
Right: graph of the potential below the half line Γ+,
1 : enter (AF ) with a slope larger
than that of Γ− by O(ℓ
2 : exit (CB) with a slope less than
that of Γ+ by O(ℓ
3 : cross (FC) and stay below (ED).
It is obvious that such conditions ensure the existence of
at least one regular point, as seen by moving (ED) down
parallel to itself until it touches the graph. Note that A
and the slope of (AB) are prescribed. Hence, one is cal-
culating conditional probabilities; but it may be shown
that the conditioning is not affecting the scaling depen-
dence on ℓ.
Φ(     )
Fig. 12. The box construction used to find a regular point R
within a Lagrangian interval of length ℓ (from [105,107]).
As the Brownian motion v0(x0) is a Markov process, the
constraints Left, Box and Right are independent and
hence,
P reg. (ℓ) ≡ Prob{regular point in interval of length ℓ}
= Prob{Left}×Prob{Box}×Prob{Right} . (3.17)
The sizes of the box were chosen so that Prob{Box} is
independent of ℓ:
Prob {Box} ∼ ℓ0. (3.18)
Indeed, Brownian motion and its integral have scaling
exponent 1/2 and 3/2, respectively, and the problem
with ℓ≪ 1 can be rescaled into that with ℓ = 1 without
changing probabilities.
It is clear by symmetry that Prob{Left} and Prob{Right}
have the same scaling in ℓ. Let us concentrate on
Prob{Right}. We can write the equation for the half
line Γ+ in the form
Γ+: x0 7→Φ(x′′0 )+δℓ3/2
(x′′0 )+γℓ
(x0−x′′0), (3.19)
where γ and δ are positive O(1) quantities. Hence, intro-
ducing α ≡ x0 − x′′0 , the condition Right can be written
to the leading order as
v0(x0) + γℓ
dx0 + δℓ
3/2 +
> 0, (3.20)
for all α > 0. By the change of variable α = βℓ and use of
the fact that the Brownian motion has scaling exponent
1/2, one can write the condition Right as
(v0(x0) + γ) dx0 > −δ, for all α ∈ [0, ℓ−1]. (3.21)
Without affecting the leading order, one can replace the
Brownian motion by a stepwise constant random walk
with jumps of ±1 at integer x0’s. The integral in (3.21)
has a simple geometric interpretation, as highlighted in
figure 13, which shows a random walk starting from
the ordinate γ and the arches determined by succes-
sive zero-passings. The areas of these arches are denoted
S⋆, S1, ...Sn, S⋆⋆.
Fig. 13. The arches construction which uses the zero-passings
of a random walk to estimate the integral of Brownian motion
(from [105,107]).
It is easily seen that
Prob{Right} ∼ Prob{S1 > 0, S1 + S2 > 0, . . .
S1 + · · · + Sn > 0 }, (3.22)
where n = O(ℓ−1/2) is the number of zero-passings of
the random walk in the interval [0, ℓ−1]. The probability
(3.22) can be evaluated by random walk methods (see,
e.g.,[49], Chap. 12, section 7), yielding
Prob{Right} ∼Prob{n first sums>0}
∝ n−1/2 ∝ ℓ1/4. (3.23)
By (3.17), (3.18) and (3.23), the probability to have a
regular point in a small interval of length ℓ behaves as
ℓ1/2 when ℓ → 0. Thus, the regular points have a box-
counting dimension 1/2.
This rigorous result on the fractal dimension of regular
points served as a basis in [4] for a proof of the bifrac-
tality of the inverse Lagrangian map when the initial
velocity is Brownian. Namely, the moments Mq(ℓ) =
〈(x0(x+ ℓ) − x0(x))〉 behave as ℓτq at small separation
ℓ and the exponents τq experience the phase transition
τq = 2q for q ≤ 1/2 (3.24)
τq = 1 for q ≥ 1/2 (3.25)
At the moment, this is the only rigorous result on the
bifractal nature of the solutions to the Burgers equation
in the case of non-differentiable initial velocity. In par-
ticular, the case of fractional Brownian motion is still
opened.
4 Transport of mass in the Burgers/adhesion
model
In the cosmological application of the Burgers equation,
i.e. for the adhesion model, it is of particular interest
to analyze the behavior of the density of matter, since
the large-scale structures are characterized as regions
where mass is concentrated. This is done by associat-
ing to the velocity field ~v solution to the d-dimensional
decaying Burgers equation (3.1), a continuity equation
for the transport of a mass density field ρ. In Eulerian
coordinates, the mass density ρ satisfies
∂tρ+ ∇ · (ρ~v) = 0 , ρ(~x, 0) = ρ0(~x) . (4.1)
A straightforward consequence of (4.1) and of the for-
mulation of Burgers dynamics in terms of characteristics
~X(~x0, t) is that, at the Eulerian locations where the La-
grangian map is invertible, the mass density field ρ can
be expressed as
ρ(~x, t)=
ρ0(~x0)
J(~x0, t)
, where ~X(~x0, t)=~x,
and J(~x0, t)=det
(∂X i)/(∂x
. (4.2)
Large but finite values of the density will be reached
at locations where the Jacobian J of the Lagrangian
map becomes very small. As we will see in section 4.1,
they contribute a power-law behavior in the tail of the
probability density function of ρ.
The expression (4.2) is no more valid when the Jacobian
vanishes (inside shocks). Then the density field becomes
infinite and mass accumulates on the shock. We will see
in section 4.2 that the evolution of the mass inside the
singularities of the solution can be obtained as the ν →
0 limit of the well-posed viscous problem. Finally, we
will discuss in section 4.3 some of the applications of
the Burgers equation to cosmology, and in particular
how, assuming the dynamics of the adhesion model, the
question of reconstruction of the early Universe from
its present state can be interpreted as a convex optimal
mass transportation problem.
4.1 Mass density and singularities
We give here the proof reported in [54] that in any dimen-
sion large densities are localized near “kurtoparabolic”
singularities residing on space-time manifolds of co-
dimension two. In any dimension, such singularities con-
tribute universal power-law tails with exponent −7/2
to the mass density probability density function (PDF)
p(ρ), provided that the initial conditions are smooth.
In one dimension, the mass density at regular points can
be written as
ρ(X(x0, t), t) =
ρ0(x0)
1 − t[(d2Ψ0)/(dx20)]
. (4.3)
We suppose here that the initial density ρ0 is strictly
positive and that both ρ0 and Ψ0 are sufficiently regu-
lar statistically homogeneous random fields. Large val-
ues of ρ(x, t) are obtained in the neighborhood of La-
grangian positions with a vanishing Jacobian, i.e. where
d2Ψ0(x0)/dx
0 = 1/t. Once mature shocks have formed,
the Lagrangian points with vanishing Jacobian are in-
side shock intervals and thus not regular. The only points
with a vanishing Jacobian that are at the boundary of
the regular points are obtained at the preshocks, that is
when a new shock is just born at some time t⋆. Such
points, that we denote by x⋆0, are local minima of the ini-
tial velocity gradient which have to be negative, so that
the following relations are satisfied:
(x⋆0) =
(x⋆0) = 0,
(x⋆0) < 0 . (4.4)
There is of course an extra global regularity condition
that the preshock Lagrangian location x⋆0 has not been
captured by a mature shock at a time previous to t⋆.
This global condition affects only constants but not the
scaling behavior of p(ρ) at large ρ.
We now Taylor-expand the initial density and the initial
velocity potential in the vicinity of x⋆0. By adding a suit-
able constant to the initial potential, shifting x⋆0 to the
origin and making a Galilean transformation canceling
the initial velocity at x⋆0, we obtain the following “nor-
mal forms” for the Lagrangian potential (3.5) and for
the density
Φ(x0, t)≃
τx20+ζx
0, ρ(X(x0, t), t)≃
τ+12ζx20
, (4.5)
where
t− t⋆
and ζ =
< 0 . (4.6)
The Lagrangian potential bifurcates from a situation
where it has a single maximum at τ < 0 through a de-
generate maximum with quartic behavior at τ = 0, to
a situation where convexity is lost and where it has two
maxima at x±0 = ±
−τ/(4ζ) for τ > 0. As a result of
our choice of coordinates, the symmetry implies that the
convex hull contains a horizontal segment joining these
two maxima (see. figure 14(a)).
 , t)
τ < 0
τ = 0
τ > 0
Fig. 14. Normal form of the Lagrangian potential. (a) in
one dimension, in the time-neighborhood of a preshock; at
the time of the preshock (τ = 0), the Lagrangian potential
changes from a single extremum to three extrema and devel-
ops a non-trivial convex hull (shown as a dashed line). (b)
in two dimension, the space neighborhood of a shock ending
point has a structure similar to the spatio-temporal normal
form of a preshock in one dimension when replacing the x0,2
variable by the time τ ; the continuous line is the separatrix
between the regular part and the ruled surface of the convex
hull; the dotted line corresponds to the locations where the
Jacobian of the Lagrangian map vanishes.
We see from (4.5) that the Eulerian density ρ is propor-
tional to x20 in Lagrangian coordinates at t = t⋆. Since
X = −∂x0Φ, the relation between Lagrangian and Eu-
lerian coordinates is cubic, so that at τ = 0, the den-
sity has a singularity ∝ x−2/3 in Eulerian coordinates.
At any time t 6= t⋆, the density remains bounded ex-
cept at the shock position. Before the preshock (τ < 0),
it is clear that ρ < −ρ0/τ , while after (τ > 0), exclu-
sion of the Lagrangian shock interval [x−0 , x
0 ] implies
that ρ < ρ0/(2τ). Clearly, large densities are obtained
only in the immediate space-time neighborhood of the
preshock. More precisely, it follows from (4.5) that hav-
ing ρ(x, t) > µ requires simultaneously
|τ | < ρ0
and |x| < (−12ζ)−1/2
. (4.7)
The tail of the cumulative probability of the density can
be determined from the fraction of Eulerian space-time
where ρ exceeds a given value. This leads to
P>(µ; x, t) = Prob{ρ(x, t)>µ} ≃ C(x, t)µ−5/2, (4.8)
where the constant C can be expressed as
C(x, t) = At
|ζ|−1/2p3(x, t, ζ) dζ, (4.9)
A is a positive numerical constant and p3 designates the
joint probability distribution of the preshock space-time
position and of its “strength” coefficient ζ (see [54] for
details). This algebraic law for the cumulative probabil-
ity implies that the PDF of the mass density has a power-
law tail with exponent −7/2 at large values. Actually
this law was first proposed in [37] for the large-negative
tail of velocity gradients in one-dimensional forced Burg-
ers turbulence, a subject to which we shall come back in
section 7.
In higher dimensions it was shown in [54] that the main
contribution to the probability distribution tail of the
mass density does not stem from preshocks but from
“kurtoparabolic” points. Such singularities (called A3
according to the classification of [62], which is summa-
rized in section 2.3) correspond to locations which be-
long to the regular part of the convex hull of the La-
grangian potential Φ(~x0, t) and where its Hessian van-
ishes. The name kurtoparabolic comes from the Greek
“kurtos” meaning “convex”. These points are located
on the spatial boundaries of shocks and generically form
space-time manifolds of co-dimension 2 (persisting iso-
lated points for d = 2, lines for d = 3, etc.). As in
one dimension, the normal form of such singularities is
obtained by Taylor-expanding in a suitable coordinate
frame the Lagrangian potential to the relevant order
Φ(~x0, t) ≃ ζx40,1+
2≤j≤d
x20,j +βjx
0,1x0,j
, (4.10)
where the different coefficients satisfy inequalities that
ensure that the surface is below its tangent plane at
~x0 = 0. The typical shape of the Lagrangian potential in
two dimensions is shown in figure 14(b). The positions
where the Jacobian of the Lagrangian map vanishes can
be easily determined from this normal form. The convex
hull of Φ and the area where the mass density exceeds the
value µ can also be constructed explicitly. An important
observation is that, in any dimension, the scalar product
of the vector ~y0 = (x0,2, . . . , x0,d) with the vector ~β =
(β2, . . . , βd) plays locally the same role as time does in
the analysis of one-dimensional preshocks.
When µ → ∞, the cumulative probability can be esti-
mated as
P>(µ; x, t) ∝ µ−3/2
︸ ︷︷ ︸
from x0,1
× µ−1
from ~β·~y0
× 1 × · · · × 1
︸ ︷︷ ︸
from other components of ~y0
from time
. (4.11)
The only non-trivial contributions come from x0,1 and
from the component of ~y0 along the direction of ~β, all
the other components and time contributing order-unity
factors. Hence, the cumulative probability P>(µ) is pro-
portional to µ−5/2 in any dimension, so that the PDF of
mass density has a power-law behavior with the univer-
sal exponent −7/2.
As we have seen, the theory is not very different in one
and higher dimension even if kurtoparabolic points are
persistent only in the latter case. This is due to the pres-
ence of a time-like direction in the case d ≥ 2.
4.2 Evolution of matter inside shocks
As we have seen in the previous subsection, the mass
density becomes very large in the neighborhood of kur-
toparabolic points (A3 singularities) corresponding to
the space-time boundaries of shocks. Such singularities
dominate the tail of the mass density probability dis-
tribution and contribute a power-law behavior with ex-
ponent −7/2. However the mass distribution depends
strongly on what happens inside the shocks where the
density is infinite. Indeed, after the formation of the first
singularity a finite fraction of the initial mass gets con-
centrated inside these low-dimensional structures. De-
scribing the mass distribution requires understanding
how matter evolves once concentrated in the shocks. But
before it will be useful to explain briefly the time evolu-
tion of the shock manifold.
4.2.1 Dynamics of singularities
Suppose that ~X(t) denotes the position of a shock at
time t. We suppose this singularity to be of type An1
(see section 2.3), so that at this position, the veloc-
ity field is discontinuous; we denote by ~v1, . . . , ~vn the n
different limiting values it takes at that point. At any
time we generically have n ≤ d + 1 and occasionally
n = d+2 at the space-time positions of shock metamor-
phoses corresponding to instants when two Ad1 singular-
ities merge. We first restrict ourselves to persistent sin-
gularities, meaning that n ≤ d+ 1. In the neighborhood
of ~X(t), it is easily checked that the velocity potential
can be written as
Ψ(~x, t) = Ψ( ~X(t), t) + max
j=1..n
~vj · ( ~X(t) − ~x)
+o(‖~x− ~X(t)‖) . (4.12)
This expansion divides locally the physical space in n
subdomains Ωj where ~vj · ( ~X(t) − ~x) is maximum, i.e.
~y ∈ Ωj ⇔ (~vi − ~vj) · (~y− ~X(t)) ≥ 0, 1 ≤ i ≤ n . (4.13)
Writing the expansion (4.12) amounts to approximating
the velocity potential by a continuous function which is
piecewise linear on the subdomains Ωj . The boundaries
between the Ωj ’s define the local shock manifold. The
maximum in (4.12) ensures that we are focusing on en-
tropic solutions to the Burgers equation (solutions ob-
tained in the limit of vanishing viscosity) and results in
the convexity of the local approximation of the poten-
tial. Note also that the position ~x = ~X(t) of the reference
singular point corresponds by construction to the unique
intersection of all subdomains Ωj . Remember that we
have assumed that locally, the solution does not have
higher-order singularity.
The approximation (4.12) fully describes the local struc-
ture of the singularity. If n = 2, corresponding to ~X(t)
being the position of a simple shock, it is easily checked
from (4.12) that there will actually exist a whole shock
hyper-plane given by the set of positions ~y satisfying
(~v1 − ~v2) · ( ~X(t) − ~y) = 0 . (4.14)
If n > 2, meaning that ~X(t) is an intersection be-
tween different shocks, all the singular manifolds of
co-dimension m ≤ n are present in the expansion and
are given by the set of positions ~y satisfying
~vi1 · ( ~X(t) − ~y) = · · · = ~vim · ( ~X(t) − ~y) , (4.15)
with 1 ≤ i1 < · · · < im ≤ n.
We next apply the variational principle (3.2) in order to
solve the decaying problem between times t and t + δt
with the initial condition given by (4.12). This yields an
approximation of the potential at time t+ δt:
Ψ(~x, t+ δt) ≃ Ψ( ~X(t), t)
+ max
j=1..n
~vj · ( ~X(t) − ~y) −
‖~x− ~y‖2
. (4.16)
Note that here, δt and ‖~x− ~X(t)‖ are chosen sufficiently
small in a suitable way to ensure that (i) any singularity
of higher co-dimension does not interfere with the posi-
tion of ~X(t) between times t and t+ δt and that (ii) the
subleading terms are always dominated by the kinetic
energy contribution ‖~x− ~y‖2/(2δt).
The two maxima in ~y and in j of (4.16) can be inter-
changed, under the condition that the maximum in ~y is
restricted to the domain Ωj defined in (4.13). The po-
tential at time t+ δt can thus be written as
Ψ(~x, t+ δt) ≃ Ψ( ~X(t), t)
+ max
j=1..n
~y∈Ωj
~vj · ( ~X(t) − ~y) −
‖~x− ~y‖2
. (4.17)
We next remark that for all ~x, j and ~y, one has
~vj · ( ~X(t) − ~y) −
‖~x− ~y‖2
≤ ~vj · ( ~X(t) − ~x+ δt~vj) −
‖~vj‖2 ,(4.18)
which gives an upper-bound to the maximum over ~y ∈
Ωj in (4.17). Suppose now that the maximum over the
index j is achieved for j = j0. This means that for all
1 ≤ i ≤ n and ~y ∈ Ωi
~vi · ( ~X(t) − ~y) −
‖~x− ~y‖2
≤ max
~z∈Ωj0
~vj0 · ( ~X(t) − ~z) −
‖~x− ~z‖2
≤ ~vj0 · ( ~X(t) − ~x+ δt~vj0) −
‖~vj‖2 . (4.19)
Let Ωi0 be the domain containing the vector (~x− δt~vj0).
Then, (4.19) applied to i = i0 and ~y = ~x− δt~vj0 trivially
implies that
(~vi0 − ~vj0 ) · (~x− δt~vj0 − ~X(t)) ≥ 0 , (4.20)
which together with the definition (4.13) for Ωi0 leads
to i0 = j0. Hence, to summarize, if the first maximum
is reached for j = j0 then the second maximum is nec-
essarily reached for ~y = ~x− δt~vj0 .
It is clear that the approximation (4.16) of the velocity
potential at time t + δt preserves the local structure of
the singular manifold. Indeed, for m ≤ n, the positions
~y satisfying
~v1 · ( ~X(t) − ~y) +
‖~v1‖2 = · · ·
· · ·= ~vm · ( ~X(t) − ~y) +
‖~vm‖2 (4.21)
form a (d−m)-dimensional shock manifold. The trajec-
tory ~X(t) of the reference singular point satisfies
~v1 ·
‖~v1‖2 = · · · = ~vn ·
‖~vn‖2 , (4.22)
which can be rewritten as
‖d ~X/dt− ~v1‖ = · · · = ‖d ~X/dt− ~vn‖ . (4.23)
This gives n relations for the d components of the vector
d ~X/dt. These relations allow determining the normal
velocity of the singular manifold. The tangent velocity
remains undetermined. The velocity of the singularity
located at ~X(t) is completely determined only if n = d,
i.e. for point singularities. For instance when d = 1, the
dynamics of shocks is given by
(u1 + u2) , (4.24)
meaning that they move with a velocity equal to the half
sum of their right and left velocities. For d = 2, only the
positions of triple points (singularities of type A31 corre-
sponding to the intersection of three shock lines) are well
determined. It is easily checked that the two-dimensional
velocity vector d ~X/dt is the circumcenter of the trian-
gle formed by the three limiting values (~v1, ~v2, ~v3) that
are achieved by the velocity field at this position (see
figure 15).
v 3v node
v mass
Fig. 15. Determination of the velocity of a triple point and
of that of the mass inside it when the three limiting values
of the velocity ~v1, ~v2, and ~v3 form an obtuse triangle. The
dash-dotted circle is the circumcircle whose center gives the
velocity of the singularity and the dashed circle is the small-
est circle containing the triangle whose center gives the ve-
locity of mass.
4.2.2 Dynamics of the mass inside the singular mani-
One of the central themes of this review article is a con-
nection between Lagrangian particle dynamics and the
inviscid Burgers equation. In the unforced case the ve-
locity is conserved along particle trajectories minimizing
the Lagrangian action (see section 2). At a given mo-
ment of time, all particles whose trajectories are not min-
imizers have been absorbed by the shocks. In the one-
dimensional case when shocks are isolated points, par-
ticles absorbed by shocks just follow the dynamics of a
shock point. However, in the multi-dimensional case the
geometry of the singular shock manifold can be rather
complicated. This results in a non-trivial particle dy-
namics inside the singular manifold. In other words, the
particle absorbed by shocks have a rich afterlife and
the main problem is to describe their dynamical proper-
ties inside the singular manifold. This problem was ad-
dressed by I. Bogaevsky in [18].
The basic idea is to consider first particle transport by
the velocity field given by smooth solutions to the viscous
Burgers equation. Indeed, let ~vν(x, t) be a solution to
the viscous Burgers equation
ν + (~vν · ∇)~vν = ν∇2~vν −∇F (~x, t).
Then the dynamics of a Lagrangian particle labeled by
its position ~x0 at time t = 0 is described by the system
of ordinary differential equations
~̇Xν(~x0, t) = ~v
ν( ~Xν(~x0, t), t), ~X
ν(~x0, 0) = ~x0, (4.25)
where the dots stand for time derivatives. It is possible to
show that in the inviscid limit ν → 0 solutions to (4.25)
converge to limiting trajectories { ~X(~x0, t)}. These lim-
iting trajectories are not disjoint anymore. In fact, two
trajectories corresponding to different initial positions
~x10 and ~x
0 can merge:
~X(~x10, t
∗) = ~X(~x20, t
∗). This corre-
sponds to absorption of particles by the shock manifold.
Of course, two trajectories coincide after they merge:
~X(~x10, t) =
~X(~x20, t) for t ≥ t∗. Particles which until time
t never merged with any other particles correspond to
minimizers. Such trajectories obviously satisfy the lim-
iting differential equation:
~̇X(~x0, t) = ~v( ~X(~x0, t), t), ~X(~x0, 0) = ~x0, (4.26)
where ~v(x, t) is the entropic solution of the inviscid Burg-
ers equation which is well defined outside of the shock
manifold. However, we are mostly interested in the dy-
namics of particles which have merged with other par-
ticles and thus were absorbed by shocks. One can prove
that for such trajectories one-sided time derivatives exist
~X(t) = lim
∆t→0+
~X(t+ ∆t) − ~X(t)
(4.27)
and satisfy a “one-sided” differential equation:
~X(t) = ~v(s)( ~X(t), t). (4.28)
Here ~v(s)(·, t) is the velocity field on the shock manifold
(index s stands for shocks). It turns out that ~v(s)(~x, t)
and the corresponding shock trajectories satisfy a vari-
ational principle, described hereafter. Denote by Ψ(~x, t)
a potential of the viscous velocity field ~v(~x, t): ~v(~x, t) =
−∇Ψ(~x, t). As we have pointed out many times be-
fore,−Ψ(~x, t) corresponds to a minimum Lagrangian ac-
tion among all the Lagrangian trajectories which pass
through point ~x at time t. Shocks correspond to a situa-
tion where the minimum is attained for several different
trajectories. Correspondingly, one has several smooth
branches such that Ψ(~x, t) = Ψi(~x, t), 1 ≤ i ≤ k. Sup-
pose a particle moves from a point of shock (~x, t) with
a velocity ~v. Then at infinitesimally close time t+ δt its
position will be ~x+~vδt. In linear approximation (see pre-
vious subsection) the Lagrangian action of this infinites-
imal piece of trajectory is equal to [|~v|2/2 − F (~x, t)]δt.
Of course, the action minimizing trajectory at the point
(~x + ~vδt, t + δt) does not pass through a shock point
(~x, t). Hence, the minimum action −Ψ(~x + ~vδt, t + δt)
is smaller than −Ψ(~x, t) + [‖~v‖2/2 − F (~x, t)]δt for any
velocity ~v. However, we can put a variational condition
on ~v which requires the difference between −Ψ(~x, t) +
[‖~v‖2/2 − F (~x, t)]δt and −Ψ(~x + ~vδt, t + δt) to be as
small as possible. This is exactly the variational principle
which determines the velocity ~v = ~v(s)(~x, t) at a shock
point. It is easy to see that in linear approximation
Ψ(~x+~vδt, t+δt) = max
1≤i≤k
[Ψi(~x+~vδt, t+δt)]
= Ψ(~x, t) − min
1≤i≤k
[−∇Ψi(~x, t) · ~v − ∂tΨi(~x, t)] δt. (4.29)
Let us denote by ~vi the limiting velocities −∇Ψi(~x, t)
at the shock point (~x, t). Then, using Hamilton–Jacobi
equation for the velocity potential
∂tΨi(~x, t) =
‖∇Ψi(~x, t)‖2 + F (~x, t)
‖~vi‖2 + F (~x, t) (4.30)
we have
Ψ(~x+ ~vδt, t+ δt) = Ψ(~x, t) −
− min
1≤i≤k
~vi ·~v −
‖~vi‖2
δt−F (~x, t)δt. (4.31)
Hence, the difference of actions can be written as
∆A=−Ψ(~x, t)+ 1
‖~v‖2δt+ Ψ(~x+~vδt, t+δt)
‖~v‖2δt− min
1≤i≤k
~vi · ~v −
‖~vi‖2
1≤i≤k
‖~v − ~vi‖2δt. (4.32)
Obviously minimization of ∆A over ~v corresponds to a
center of a minimum ball covering ~vi. It implies that
such a center gives the velocity ~v(s)(~x(t), t) of particles
concentrated at a shock point (~x, t). It is interesting that
this variational principle implies that a particle absorbed
by a shock cannot leave the singular shock manifold in
the future.
Let us now consider the first nontrivial generic exam-
ple of a shock point, namely a triple point in two di-
mensions d = 2. The point ( ~X(t), t) is thus the intersec-
tion of three shock lines. In this case there are exactly
three smooth branches Ψi(·, t) with limiting velocities
~vi = −∇Ψi, 1 ≤ i ≤ 3. As we have seen in previous sec-
tion the motion of the triple point is determined by con-
tinuity of the velocity potential at ( ~X, t). The “geomet-
rical velocity” d ~X/dt of the triple point is then the cir-
cumcenter of the triangle formed by the three velocities
~v1, ~v2, ~v3. It is easy to see that d ~X/dt = ~v
(s) only in the
case when the vectors ~v1, ~v2, and ~v3 form an acute trian-
gle. If so, a cluster of particles follows the triple point. In
the opposite case when the triangle is obtuse, the parti-
cles leave the node. Such a situation is presented in fig-
ure 15, where the mass leaves the node along the shock
line delimiting the values ~v1 and ~v3 of the velocity.
The analysis presented above was carried out for the
Burgers equation jointly with A. Sobolevskĭı as a part of
ongoing work on a similar theory for the case of a general
Hamilton–Jacobi equation, with a Hamiltonian that is
convex in the momentum variable. The formal extension
of this analysis to the latter case is straightforward and
can be left to the interested reader; however at present a
rigorous justification of it, employing methods of [18], is
known only for the case of H(x, ẋ, t) = a(x, t)|ẋ|2, with
a(x, t) > 0.
4.3 Connections with convex optimization problems
As discussed in section 1.2, Burgers dynamics is known
in cosmology as the adhesion model and frequently used
to understand the formation of the large-scale structures
in the Universe. Recently, this model was used as a basis
for developing new techniques for one of the most chal-
lenging questions in modern cosmology, namely recon-
struction. This problem aims at reconstructing the dy-
namical history of the Universe through the mass den-
sity initial fluctuations that evolved into the distribution
of matter and galaxies which is nowadays observed (see,
e.g., [98]). The main difficulty encountered is that the
velocities of galaxies (the peculiar velocities) are usually
unknown, so that most approaches lead to non-unique
solutions to this ill-posed problem. The reconstruction
technique we present here, which was proposed in [55,25],
is based on the observation that, to the leading order,
the mass is initially uniformly distributed in space (see,
e.g., [98]). This observation, together with the Zeldovich
approximation, leads to a reformulation of the prob-
lem as a well-posed instance of an optimal mass trans-
portation problem between the initial (uniform) and the
present (observed) distributions of mass. More precisely
it amounts to a convex optimization problem related
to the Monge–Ampère equation and dually, as found
by Kantorovich [73], to a linear programming problem.
This is the reason why the name MAK (Monge–Ampère–
Kantorovich) has been proposed for this method in [55].
Namely, one has to find the transformation from initial
to current positions (the Lagrangian map) that maps
the initial density ρ(~x0, 0) = ρ0 to the field ρ(~x, t) which
is nowadays observed. One then use a well-known fact
in cosmology: because of the expansion of the Universe,
the initial velocity field of the self-gravitating matter is
slaved to the initial gravitational field (see, e.g., [25]).
This observation implies that the initial velocity field is
potential and allows one to deduce from it the sublead-
ing fluctuations of the mass density.
The MAK reconstruction technique is based on two cru-
cial assumptions. First the Lagrangian map ~x0 7→ ~x =
~X(~x0, t) is assumed to be potential, i.e. ~X = ∇x0Φ(~x0).
Second, the Lagrangian potential Φ(~x0) is assumed to
be a convex function. As explained in [25] these two hy-
potheses are motivated by the adhesion model (and thus
inviscid Burgers dynamics) where they are trivially sat-
isfied. As we will see later the reverse is actually true: the
potentiality of the Lagrangian map and the convexity of
the potential is equivalent to assuming that the latent ve-
locity field is a solution to the Burgers equation. We will
now see how, under these hypotheses, the reconstruction
problem relates to Monge–Ampère equation. Conserva-
tion of mass trivially implies that ρ(~x, t)d3x = ρ0d
which can be rewritten in terms of the Jacobian matrix
(∂X i)/(∂x
0) as
ρ( ~X(~x0, t), t)
. (4.33)
Potentiality of the Lagrangian map leads to
∂xi0∂x
ρ(∇x0Φ, t)
. (4.34)
The problem with this formulation is that the unknown
potential Φ enters the right-hand side of the equation
in a non-trivial way. Convexity of the Lagrangian po-
tential Φ is next used to reformulate the problem in
term of the inverse Lagrangian map. Indeed, if Φ is con-
vex, the inverse Lagrangian map is also potential, i.e.
~x0 = ~X0(x, t) = ∇xΘ(~x) with the potential Θ itself con-
vex. The two potentials Φ and Θ are moreover related
by Legendre transforms:
Θ(~x) = max
[~x · ~x0 − Φ(~x0)], (4.35)
Φ(~x0) = max
[~x · ~x0 − Θ(~x)]. (4.36)
In terms of the inverse Lagrangian potential Θ the con-
servation of mass (4.34) reads
∂xi∂xj
= ρ(~x, t), (4.37)
which is exactly the elliptic Monge-Ampère equation.
This time, the difficulty expressed above has disappeared
since the unknown potential Θ does not enter the right-
hand side of the equation. Note that we have implic-
itly assumed here that the present distribution of mass
has no singularity. The case of a singular distribution
could actually be treated using a weak formulation of
the Monge-Ampère equation, which amounts to apply-
ing conservation of mass on any subdomain but requires
allowing the inverse Lagrangian map to be multival-
ued. The next step in the design of the MAK method is
to reformulate (4.37) as an optimal transport problem
with quadratic cost. Indeed, as shown in [24], the map
~X(~x0, t) (and its inverse ~X0(~x, t)) minimizing the cost
‖ ~X(~x0, t) − ~x0‖2ρ0 d3x0
‖~x− ~X0(~x, t)‖2ρ(~x, t) d3x, (4.38)
is a potential map whose potential is convex and is the
solution to the Monge–Ampère equation (4.37). This can
be understood using a variational approach as proposed
in [55]. Suppose we perform a small displacement δ ~X0(~x)
of the inverse Lagrangian map ~X0(~x, t) solution of the
optimal transport problem. On the one hand the only ad-
missible displacement are those satisfying the constraint
to map the initial density field ρ0 to the final one ρ(~x, t).
It is shown in [25] that this is equivalent to require that
∇x · [ρ(~x, t)δ ~X0(~x)] = 0. On the other hand one easily
see that the variation of the cost function corresponding
to the variation δx reads
δI = −2
[~x− ~X0(~x, t)] · [ρ(~x, t)δ ~X0(~x)] d3x. (4.39)
This integral can be interpreted as the scalar product
(in the L2 sense) between ~x− ~X0(~x0, t) and ρ(x)δ ~X(~x0).
Hence the optimal solution, which should satisfy δI = 0
for all δ ~X0, is such that the displacement ~x − ~X0(~x0, t)
(or equivalently ~X(~x0) − ~x0) is orthogonal to all
divergence-free vector fields. This means that it is nec-
essarily the gradient of a potential, from which it follows
that ~X(~x0, t) = ∇x0Φ(~x0). Convexity follows from the
observation that the Lagrangian map ~x0 7→ ~X has to
satisfy
(~x0 − ~x′0) · [ ~X(~x0) − ~X(~x′0)] ≥ 0. (4.40)
Indeed, if that was not the case, one can easily check that
any map where the Lagrangian pre-image of a neighbor-
hood of ~x0 and of one of ~x0
are inverted would lead to a
smaller cost. Formulated in terms of potential maps, the
relation (4.40) straightforwardly implies convexity of Φ.
This finishes the proof of equivalence between Monge–
Ampère equation and the optimal transport problem
with quadratic cost.
The goal of reformulating reconstruction as an optimiza-
tion problem is mostly algorithmic. Once discretized,
the problem of finding the optimal map between initial
and final positions amounts is equivalent to solving a so-
called assignment problem. An efficient method to deal
numerically with such problems is based on the auction
algorithm [15] and was used in [25] with data stemming
from N -body cosmological simulations. As summarized
0.2 0.3 0.7 0.80.4 0.5 0.6
simulation coordinate
1 2 3 4
distances
Fig. 16. Test of the MAK reconstruction for a sample of
N ′ = 17, 178 points from a N-body simulation (from [25]).
The scatter diagram plots reconstructed versus true initial
positions. The histogram inset gives the distribution (in per-
centages) of distances between true and reconstructed initial
positions; the horizontal unit is the distance between two
sampled points. The width of the first bin is less than unity to
ensure that only exactly reconstructed points fall in it. More
than sixty percent of the points are exactly reconstructed.
in figure 16, the MAK reconstruction method leads to
very promising results. More than 60% of the discrete
points are assigned to their actual Lagrangianpre-image.
Such a number has to be compared with other recon-
struction methods for which the success rate barely ex-
ceed 40% for the same data set.
Even if the mapping from initial to final positions is
unique, the peculiar velocities are not well defined ex-
cept if we have some extra knowledge of what is happen-
ing at intermediate times 0 ≤ t′ ≤ t. Of course the den-
sity field ρ(~x′, t′) is unknown. However, there are triv-
ial physical requirements. First the two mass transport
problems between 0 and t′ and between t′ and t have
both to be optimal. This means that one looks for two
Lagrangian maps, ~X1 from 0 to t
′ and ~X2 from t
′ to t
which are minimizing the respective costs
‖ ~X1(~x0) − ~x0‖2ρ0 d3x0,
‖~x− ~X−12 (~x)‖
2ρ(~x, t) d3x. (4.41)
The second physical requirement is that the composition
of these two optimal maps have to give the Lagrangian
map between times 0 and t, namely ~X(~x0, t) = ~X2( ~X1).
Under these two conditions there is equivalence between
the optimal transport with a quadratic cost and the
Burgers dynamics supplemented by the transport of a
density field (see [13] for details).
5 Forced Burgers turbulence
5.1 Stationary régime and global minimizer
We consider in this section solutions to the forced Burg-
ers equation. As we have seen in section 2, the solution in
the limit of vanishing viscosity can be expressed at any
time t in terms of the initial condition at time t0 through
a variational principle which consists in minimizing an
action along particle trajectories. The statistically sta-
tionary régime toward which the solution converges at
large time can be studied assuming that the by reject-
ing the initial time t0 is at minus infinity. The solution
is then given by the variational principle
Ψ(~x, t)=−inf
~γ(·)
‖~̇γ(s)‖2−F (~γ(s), s)
, (5.1)
where the infimum is taken over all (absolutely continu-
ous) curves ~γ : (−∞, t] → Ω such that ~γ(t) = ~x. In this
setting, the action is computed over the whole half line
(−∞, t] and the argument of the infimum does not de-
pend anymore on the initial condition. Of course, (5.1)
defines Ψ up to an additive constant. This means that
only the differences Ψ(~x, t)−Ψ(0, t) can actually be de-
fined. A trajectory ~γ minimizing (5.1) is called a one-
sided minimizer. It is easily seen from (5.1) that all the
minimizers are solutions of the Euler–Lagrange equation
~̈γ(s) = −∇F ( ~γ(s), s) , (5.2)
where the dots denote time derivatives. This equation
defines a 2d-dimensional (possibly random) dynamical
system in the position-velocity phase space (~γ, ~̇γ). The
Lagrangian one-sided minimizers ~γ defined over the half-
infinite interval (−∞, t] play a crucial role in the con-
struction of the global solution and of the stationary
régime. Namely, a global solution to the randomly forced
inviscid Burgers equation is given by~v(~x, t) = ~̇γ(t) where
~γ(t) = ~x. To prove that such half-infinite minimizers ex-
ist, one has to take the limit t0 → −∞ for minimizers
defined on the finite time interval [t0, t]. The existence of
this limit follows from a uniform bound on the absolute
value of the velocity |~̇γ| (see, e.g., [38]). Obtaining such
a bound becomes the central problem for the theory, as
we shall now see.
When the configuration space Ω where the solutions live
is compact (bounded), one can expect the velocity of a
minimizer to be uniformly bounded. Indeed, in this case
the displacement of a minimizer for any time interval is
then bounded by the diameter of the domain Ω, so that
action minimizing trajectories cannot have large veloc-
ities. For forcing potential that are delta-correlated in
time, it has been shown by E et al. [38] in one dimension
and by Iturriaga and Khanin [68,69] in higher dimen-
sions that the minimizing problem (5.1) has a unique
solution Ψ with the following properties:
• Ψ is the unique statistically stationary solution to the
Hamilton–Jacobi equation (2.2) in the inviscid limit
ν → 0;
• Ψ is almost everywhere differentiable with respect to
the space variable ~x;
• −∇Ψ uniquely defines a statistically stationary solu-
tion to the Burgers equation in the inviscid limit;
• there exists a unique one-sided minimizer at those Eu-
lerian positions ~x where the potential Ψ is differen-
tiable; the locations where Ψ is not differentiable cor-
respond to shocks.
• There exists a unique minimizer ~γ(g) that minimizes
the action calculated from −∞ to any time t. It is
called the global minimizer (or two-sided minimizer)
and corresponds to the trajectory of a fluid particle
that is never absorbed by shocks. Moreover, all one-
sided minimizers are asymptotic to it as s→ −∞.
All the properties above follow from the variational ap-
proach. In fact, the variational principle (2.12) imply
similar statements in the viscous case. Of course, when
viscosity is positive the unique statistically stationary
solution is smooth. However, one can show that the sta-
tionary distribution corresponding to such solutions con-
verges to inviscid stationary distribution in the limit
ν → 0 [58]. Although the variational proofs are concep-
tual, general and simple, they are based on the fluctua-
tion mechanism and therefore do not give a good control
of the rate of convergence to the statistically stationary
regime. Exponential convergence would follow from the
hyperbolicity of the global minimizer. Although one ex-
pects hyperbolicity holds in any dimension, mathemat-
ically it is an open problem. At present a rigorous proof
of hyperbolicity is only available in dimension one [38].
The assumption of compactness of the configuration
space Ω is essential in the construction of the stationary
régime. As we will see in subsection 5.4, the situation
is much more complex in the non-compact case when
for instance the solution is defined on the whole space
Ω = Rd.
5.2 Topological shocks
To introduce the notion of topological shock we first fo-
cus on the one-dimensional case in a periodic domain,
i.e. in Ω = T = R/Z. If we “unwrap” at a given time t
the configuration space to its universal cover R (see fig-
ure 17(a)), we then obtain an infinite number of global
minimizer γ
k , which at all time s ≤ t satisfy γ
k+1(s) =
k (s) + 1. All the one-sided minimizers converge back-
ward in time to one of these global minimizers. The topo-
logical shock (or main shock) is defined as the set of x
positions giving rise to several minimizers approaching
two successive replicas of the global minimizer. This par-
ticular shock is also the only shock that has existed for
all times.
This construction can easily be extended to higher
dimensions (see [10]). For this we unwrap the d-
dimensional torus Td to its universal cover, the full
space Rd (see figure 17(b) for d = 2). Then, the different
replicas of the periodic domain define a lattice of global
minimizers ~γ
parameterized by integer vectors ~k. The
backward-in-time convergence on the torus of the one-
sided minimizers to the global minimizer implies that
a minimizer associated to a location ~x in Rd at time t
will be asymptotic to one of the global minimizer ~γ
the lattice. Hence, every position ~x which has a unique
one-sided minimizer is associated to an integer vector
~k(~x). This defines a tiling of space at time t. The tiles
O~k are the sets of points whose associated one-sided
minimizers are asymptotic to the ~k-th global minimizer.
The boundaries of the O~k’s are the topological shocks.
They are the locations from which at least two one-
sided minimizers approach different global minimizers
on the lattice. Indeed, a point where two tiles O~k1 and
O~k2 meet, has at least two one-sided minimizers, one
of which is asymptotic to ~γ
and another to ~γ
course, there are also points on the boundaries where
three or more tiles meet and thus where more than two
one-sided minimizers are asymptotic to different global
minimizers. For d = 2 such locations are generically iso-
lated points corresponding to the intersections of three
or more topological shock lines, while for d = 3, they
form edges and vertices where shock surfaces meet. Note
Fig. 17. Space-time sketch of the unwraping of the periodic
domain Td to the whole space Rd for d = 1 (a) and d = 2 (b).
that, generically, there exist other points inside O~k with
several minimizers. They correspond to shocks of “lo-
cal” nature because at these locations, all the one-sided
minimizers are asymptotic to the same global minimizer
and hence, to each other. In terms of Lagrangian
dynamics, the topological shocks play a role dual to that
of the global minimizer. Indeed, all the fluid particles
are converging backward-in-time to the global mini-
mizer and are absorbed forward-in-time by the topo-
logical shocks. For the transportation of mass when we
assume that the Burgers equation is supplemented by
a continuity equation for the mass density, all the mass
concentrate at large times in the topological shocks.
The global structure of the topological shocks is related
to the various singularities generically present in the so-
lution to the Burgers equation that were detailed in sec-
tion 2.3. Generically there are no locations associated to
more than (d+1) minimizers. As one expects to see only
generic behavior in a random situation, the probability
to have points with more than (d + 1) one-sided mini-
mizers is zero. It follows that there are no points where
(d+ 2) tiles O~k meet, which is an important restriction
on the structure of the tiling. For d = 2 it implies that
the tiling is constituted of curvilinear hexagons. Indeed,
suppose each tile O~k is a curvilinear polygon with s ver-
Fig. 18. (a) Position of the topological shock on the torus;
the two triple points are represented as dots. (b) Snapshot
of the velocity potential ψ(x, y, t) for d = 2 in the statisti-
cal steady state, obtained numerically with 2562 grid points.
Shock lines, corresponding to locations where ψ is not dif-
ferentiable, are represented as black lines on the bottom of
the picture; the four gray areas are different tiles separated
by the topological shocks; the other lines are local shocks.
tices corresponding to triple points. For a large piece
of the tiling that consists of N tiles, the total num-
ber of vertices is nv ∼ sN/3 and the total number of
edges is ne ∼ sN/2. The Euler formula implies that
1 = nv − ne +N ∼ (6− s)N/6, and we necessarily have
s = 6, corresponding to an hexagonal tiling. As shown
in figure 18(a), this structure corresponds on the peri-
odicity torus T2, to two triple points connected by three
shock lines that are the curvilinear edges of the hexagon
O~0. The connection between the steady-state potential
and the topological shocks is illustrated numerically on
figure 18(b). The different tiles covering the periodic do-
main were obtained by tracking backward in time fluid
particle trajectories and by determining to which peri-
odic image of the global minimizer they converge.
In dimensions higher than two, the structure of topolog-
ical shocks is more complicated. For instance it is not
possible to determine in a unique manner the shape of
the polyhedra forming the tiling. However, it has been
shown by Matveev [87] that for d = 3 the minimum poly-
hedra forming such tiling has 24 vertices and 36 edges
and is composed of 8 hexagons and 6 rectangles (see fig-
ure 19). It is of interest to note that the structure of
topological shocks is in direct relation with the notions
of complexity and minimum spines of manifolds intro-
duced by Matveev from a purely topological viewpoint.
Fig. 19. Sketch of the simplest configuration of the topolog-
ical shock in dimension d = 3.
Algebraic characterization of the topological shock
In two dimensions, when periodic boundary conditions
are considered, very strong constraints are imposed on
the structure of the solution. In particular, the topol-
ogy of the torus T2 imply that the topological shocks
generically form a periodic tiling of R2 with curvilinear
hexagons. However, this tiling can be of various alge-
braic types. Consider the tile O~0 surrounded by its six
immediate neighbors O~ki , where the integer vectors
are labeled in anti-clockwise order, ~k1 having the small-
est polar angle (see figure 20). It is easily seen that the
periodicity of the tiling implies
~k3 = ~k2 − ~k1, ~k4 = −~k1, ~k5 = −~k2
and ~k6 = ~k1 − ~k2, (5.3)
so that the whole information on the algebraic struc-
ture of the tiling is contained in the vectors ~k1 and ~k2
which form a matrix S from the group SL(2,Z) of 2× 2
integer matrices with unit determinant. The matrix S
gives information on the number of times each shock line
turns around the torus before reconnecting to another
triple point. Figure 18(a) corresponds to the simplest
case when S is the identity matrix. When the forcing is
stochastic, the matrix S is random and stationary solu-
tions to the two-dimensional Burgers equation define a
stationary distribution on SL(2,Z).
(i, j)
(i−1, j+1)
(i+1, j)
(i+1, j−1)
(i, j+1)
(i−1, j)
(i, j−1)
Fig. 20. The algebraic structure of the topological shock in
dimension d = 2 is determined by the indexes corresponding
to immediate neighbors of the tiling considered.
Certainly, topological shocks evolve in time and may
change their algebraic structure. This happens through
bifurcations (or metamorphoses) described in section
2.3. In two dimensions, the generic mechanism which
transforms the algebraic structure of topological shocks
is the merger of two triple points. This metamorphosis
is called the flipping bifurcation and corresponds to the
appearance at time t⋆ of an A
1 singularity in the solu-
tion associated to a position with four minimizers. The
mechanism transforming the algebraic structure of the
topological shock is illustrated in figure 21. Issues such
as the minimum number of flips needed to transform the
matrix S1 associated to the algebraic structure of the
topological shock to another matrix S2 are discussed in
in [1].
Fig. 21. Sketch of the tiling before, at the flipping time t∗ and
after it. This example corresponds to a bifurcation from the
matrix S1 = [
1] to S2 = [
2]. The dashed boxes represent
the periodicity domain [0, 1]2.
5.3 Hyperbolicity of the global minimizer
The nature of the convergence to a statistical steady
state is determined by the local properties of the global
minimizer. The hyperbolicity of this action-minimizing
trajectory implies an exponential convergence, so that
the global picture of the solution is reached very rapidly,
after just a few turnover times.
Since the trajectory of the global minimizer is unique
and can be extended to arbitrary large times, it corre-
sponds to an ergodic invariant measure for the stochastic
flow defined by the Euler–Lagrange equation (5.2). Con-
ditioned by the random force, this measure is simply the
delta measure sitting at the location (~γ(g)(0), ~̇γ(g)(0)).
By the Oseledets ergodic theorem (see, e.g. [98]), 2d non-
random Lyapunov exponents can be associated to the
global minimizer trajectory. Since the flow is symplectic
these non-random exponents come in pairs with oppo-
site signs. That is
λ1 ≥ · · · ≥ λd ≥ 0 ≥ −λd ≥ · · · ≥ −λ1 . (5.4)
Hyperbolicity is defined as the non-vanishing of all these
exponents. Thus, the issue of hyperbolicity can be ad-
dressed in terms of the backward-in-time convergence of
the one-sided minimizers to the global one or, better, in
terms of forward-in-time dynamics. In the latter case,
this amounts to looking how fast Lagrangian fluid par-
ticles are absorbed by shocks. For this we consider the
set Ωreg(T ) of locations ~x such that the fluid particle
emanating from ~x at time t = 0 survives, i.e. is not ab-
sorbed by any shock, until the time t = T . The long-time
shrinking of Ωreg as a function of time is asymptotically
governed by the Lyapunov exponents. To ensure the ab-
sence of vanishing Lyapunov exponents, it is sufficient to
show that the diameter of Ωreg(T ) decays exponentially
as T → +∞.
In one dimension, it has been shown in [38] that this is
indeed the case, and particularly that there exists posi-
tive constants α, β, A and B such that
diamΩreg(T ) ≥ Ae−αT
≤ Be−βT . (5.5)
Unfortunately this proof of hyperbolicity is purely one-
dimensional and at present time there is no extension of
this result to higher dimensions.
In two dimensions, the behavior of diamΩreg(T ) at large
times was studied numerically in [10] by using the fast
Legendre transform described in section 2.4 and a forc-
ing that is a sum of independent random impulses con-
centrated at discrete times. The ideas of this numerical
method are related to the Lagrangian structure of the
flow. This easily permits to track numerically the set Ωreg
of regular Lagrangian locations. As seen from figure 22,
the diameter of this set decays exponentially fast in time
for three different types of forcing, providing good ev-
idence of the hyperbolicity of the global minimizer for
d = 2.
0 0.25 0.5 0.75 1
time T
Algebraic  [∝ k−3]
Gaussian   [∝ exp (−k2)]
Step           [= Const. if k ≤3]
Fig. 22. Time evolution of the diameter of the Lagrangian
set Ω(T ) (points corresponding to the regular region) for
three different types of forcing spectrum; average over 100
realizations and with 2562 grid points (from [10]).
Hyperbolicity of the global minimizer implies existence
at any time t of two d-dimensional smooth manifolds
u x,t(    )
(  )tΓ(u)
global
minimizer
shock
a preshock
Fig. 23. Sketch of the unstable manifold for d = 1 in the
(x, v) plane. Shock locations (A21 singularities) are obtained
by applying Maxwell rules to the loops. A preshock (A3
singularity) is represented; it corresponds to the formation
of a loop in the manifold. The velocity profile which is the
actual solution to the Burgers equation is represented as a
bold line.
in phase space (~γ, ~̇γ) that are invariant by the Euler–
Lagrange dynamics (5.2): a stable (attracting) manifold
Γ(s)(t) and an unstable (repelling) manifold Γ(u)(t), de-
fined as the instantaneous location of trajectories con-
verging to the global minimizer forward in time and
backward in time, respectively. Since all the minimizers
converge backward in time to the global minimizer, the
graph in the position-velocity phase space (~x,~v) of the
solution in the statistical steady state is made of pieces of
the unstable manifold Γ(u)(t) with discontinuities along
the shocks lines or surfaces. In other words, shocks ap-
pear as jumps between two different folds of the unstable
manifold. The smoothness of the unstable manifold is an
important property; for instance, it implies that when
d = 2, the topological shock lines are smooth curves.
In one dimension, where hyperbolicity is ensured, the
main shock corresponds to a jump between the right
branch and the left branch of the unstable manifold. Its
position can be obtained geometrically after observing
that the area b covered by the unstable manifold, once
the latter is cut by the main shock, should be equal to
the first integral of motion which is conserved, i.e.
v(x, t) dx =
v0(x) dx . (5.6)
The other shocks (or secondary shocks that have existed
only for a finite time) cut through the double-fold loops
of the unstable manifold (see figure 23). Their locations
can be obtained by a Maxwell rule applied to those loops.
Indeed, the difference of the two areas defined by cutting
such a loop at some position x is equal to the difference
of actions of the two trajectories emanating from the up-
per and lower locations and, thus, vanishes at the shock
location. We will see in section 6 that this construction
of the solution is also valid when the forcing is periodic
in time, problem which can be related to Aubry–Mather
theory relative to commensurate-incommensurate phase
transitions.
The above geometrical construction of the solution has
much in common with that appearing in the unforced
problem. Indeed, as we have seen in section 3.1, when
F = 0 the solution can be obtained geometrically by
considering in the (~x,~v) space, the Lagrangian manifold
defined by the position and the velocity of the fluid par-
ticles at a given time. This analogy gives good ground
predicting that some universal properties associated to
the unforced problem will still hold in the forced case, as
we will indeed see in section 7. Another instance concerns
transport of mass in higher dimension. We have seen in
section 4.1 that, for the unforced case, large but finite
mass densities are localized near boundaries of shocks
(“kurtoparabolic” singularities) contributing power-law
tails with the exponent −7/2 to the probability density
function of the mass density. When a force is applied the
smoothness of the unstable manifold associated to the
global minimizer should lead to the same universal law.
5.4 The case of extended systems
So far, we have discussed the global structure of the
solution to the forced Burgers equation with periodic
boundary conditions. Is is however of physical interest
to understand instances when the size of the domain is
much larger than the typical length scale of the forcing.
In this section, we will focus on describing, in the one-
dimensional case, the singular structure of the solution
in unbounded domains. Based on the formalism of [11],
we achieve this goal by considering a spatially periodic
forcing with a characteristic scale Lf much smaller than
the system size L. More precisely, for a fixed size L we
consider the stationary régime corresponding to the limit
t → ∞ and then study the limit L → ∞ by keeping
constant the energy injection rate (i.e. the L2 norm of
the forcing grows like L).
In order to get an idea of the behavior of the solution,
the limit of infinite aspect ration L/Lf was investigated
numerically in [11]. As seen from figure 24(a) numerical
observations suggest that at any time in the statistical
steady state, the shape of the velocity profile is simi-
lar to the order-unity aspect ratio problem, duplicated
over independent intervals of sizeLf . In particular, when
tracking backward in time the trajectories of fluid par-
ticles the minimizers converge to each other in a very
non-uniform way. Figure 24(b) shows that the minimiz-
ers form different branches, which are converging to each
other backward in time; in space time a tree structure is
obtained. As shown in figure 25(a) a similar behavior is
observed for shocks.
The velocity field at a given time t, consists of smooth
pieces separated by shocks. Let us denote by {Ωj} the
set of intervals in [0, L), on which the solution u(·, t)
is smooth. The boundaries of the Ωj ’s are the shocks
positions. Each of these shocks is associated to a root-
like structure formed by the trajectories of the various
Fig. 24. (a) Upper: snapshot of the velocity field for
L = 256Lf . Lower: zoom of the field in a interval of
length 10Lf . (b) Minimizing trajectories in space time for
L = 256Lf and over a time interval of length T = 100
shocks that have merged at times less than t to form the
shock under consideration (see figure 25(a)). This root-
like structure contains the whole history of the shock
and in particular its age (i.e. the length of the deeper
branch of the root structure). Indeed, if the root has a
finite depth, the shock considered has only existed for a
finite time. A T -global shock is defined as a shock whose
associated root is deeper than −T . They can alterna-
tively be defined geometrically by considering the left-
most and the rightmost minimizer associated to it. After
tracing them backward for a sufficiently long time, these
two minimizers are getting close and eventually converge
to each other exponentially fast (see figure 24(b)). For
a T -global shock, the time when the two minimizers are
getting within a distance smaller than the forcing cor-
relation length Lf is larger than T . As we have seen in
section 5.2, the existence in one dimension of a main
X  (t)1 X  (t)2
space
Fig. 25. (a) Shock trajectories for aspect ratio L/Lf = 32
and with T = 10. The different gray areas correspond to
the space-time domains associated to the different smooth
pieces Ωj of the velocity field at time t = 0. (b) Sketch of
the space-time evolution of a given smooth piece Ωj located
between two shock trajectories X1(t) and X2(t) that merge
at time Tj .
shock in the spatially periodic situation follows from a
simple topological argument. The main shock can also
be defined as the only shock that has existed forever in
the past. It is hence infinitely old, contrary to all other
shocks, all of them being created at a finite time and
having a finite age. When the periodicity condition is
dropped, the main shock disappears and it is useful to
consider the T -global shocks that mimic the behavior of
a main shock over time scales larger than T .
One can dually define T -global minimizers. All the
smoothness intervals Ωj defined above, except that
which contains the global minimizer, will be entirely
absorbed by shocks after a sufficient time (see figure
25(b)). For each of these pieces, one can define a life-time
Tj as the time when the last fluid particle contained in
this piece at time t enters a shock. It corresponds to the
first time for which the shock located on the left of this
smooth interval at time t merges with the shock located
on the right. When the life-time of such an interval is
greater than T , the trajectory of the last surviving fluid
particle is here called a T -global minimizer. Note that,
when T → ∞, the number of T -main shocks and of
T -global minimizers is one, recovering respectively the
notions of main-shock and of two-sided minimizer.
time T
 = 64 
 = 128
 = 256
slope = −2/3 
Fig. 26. Density of T -main shocks as a function of T for three
different system sizes L/Lf = 64, 128 and 256; average over
100 realizations. Lower inset: local scaling exponent.
Hence, at a given instant t, and for any timelag T ,
the spatial domain [0, L) contains a certain number of
T -objects. We define their spatial density as being the
number of such objects, averaged with respect to the
forcing realizations, divided by the size of the domain L.
The density ρ(T ) of T -main shocks was investigated nu-
merically in [11] for the kicked case by using a two-step
method: first, the simulation was run until a large time t
for which the statistically stationary régime is reached;
secondly, each shock present at time t was tracked
backward-in-time down to the instant of its creation,
giving an easy way to characterize the density ρ(T ). It
is seen in figure 26 that, for three different aspect ratios
L/Lf , the density ρ(T ) displays a power-law behavior
ρ(T ) ∝ T−2/3 for the intermediate time asymptotics
Lf/urms ≪ T ≪ L/urms.
We now present a simple phenomenological theory aim-
ing to explain the scaling exponent 2/3. We consider the
solution at a fixed time (t = 0, for instance). Denote by
ℓ(T ) the typical spatial separation scale for two nearest
T -global shocks. Obviously, ℓ(T ) ∼ 1/ρ(T ). The mean
velocity of the spatial segment of length ℓ is given by
[y, y+ℓ]
u(x, 0) dx (5.7)
Since the expected value 〈u(x, 0)〉 = 0, and that the
integral in (5.7) is over an interval of size much larger
than the forcing correlation length, it is equivalent to
a sum of independent centered random variables and
scales as the Brownian motion. Hence, for large ℓ one
has the following asymptotics
[y, y+ℓ]
u(x, 0) dx ∼
ℓ, (5.8)
which gives bℓ ∼ ℓ−1/2 for mean velocity fluctuations.
Consider now the rightmost minimizer corresponding to
the left T -global shock and the leftmost minimizer re-
lated to the right one. Since there are no T -global shocks
in between, it follows that the two minimizers we se-
lected get close to each other backward-in-time around
times of the order of −T . This means that the backward-
in-time displacement of a spatial segment of length O(ℓ)
is itself O(ℓ) for time intervals of the order of T . The
corresponding displacement is given as the sum of two
competing behaviors: the first, which can be understood
as a drift induced by the local mean velocity bℓ, is due
to the mean velocity fluctuations and is responsible for
a displacement ∝ bℓT ; the second contribution is due to
a standard diffusive scale ∝ T 1/2 expressing the diffu-
sive behavior of the minimizing trajectories. Taking into
account both terms we obtain
ℓ ∼ B1T ℓ−1/2 +B2T 1/2, (5.9)
where B1 and B2 are numerical constants. It is easy to
see that the dominant contribution comes from the first
term. Indeed, if the second term were to dominate, then
ℓ would be much larger than T , which contradicts (5.9).
Hence, one has ℓ ∼ B1T ℓ−1/2, leading to the scaling
behavior
ℓ(T ) ∝ T 2/3, ρ(T ) ∝ T−2/3. (5.10)
As we have already discussed, T -global shocks are shocks
older than T . Denote by p(A) the probability density
function (PDF) for the age of shocks. More precisely,
p(A) is a density in the stationary régime of a probability
distribution of the age A(t) of a shock, say the nearest
to the origin. It follows from (5.10) that the probability
of shocks whose age is larger than A decays like A−2/3;
this implies the following asymptotics for the PDF p(A):
p(A) ∝ A−5/3. (5.11)
Actually, the power-law behavior of the density ρ(T ) of
T -global shocks can be interpreted in term of an inverse
cascade in the spectrum of the solution (although there
is no conserved energy-like quantity). Indeed, the fluctu-
ations (5.8) of the mean velocity suggest that, for large-
enough separations ℓ, the velocity potential increment
scales like
|ψ(x+ ℓ, t) − ψ(x, t)| ∝ ℓ1/2. (5.12)
This behavior is responsible for the presence of an inter-
mediate power-law range with exponent −2 in the spec-
trum of the velocity potential at wavenumbers smaller
than the forcing scale (see figure 27). In order to observe
Fig. 27. Spectrum 〈ψ̂2(k)〉 of the velocity potential in the
stationary régime for the aspect ratio L/Lf = 128. This
spectrum contains two power-law ranges: at wavenumbers
k ≫ L/Lf , the traditional ∝ k
−4 inertial range connected
to the presence of shocks in the solution and, for k ≪ L/Lf ,
an “inverse cascade” ∝ k−2 associated to the large-scale
fluctuations of ψ
the k−2 range at small wavenumbers, the spectrum of
the forcing potential must decay faster than k−2; other-
wise the leading behavior is non-universal but depends
on the functional form of the forcing correlation.
The one-dimensional randomly forced Burgers equation
in an unbounded domain has been studied in [66] with
a different type of forcing: it was assumed that the forc-
ing potential has at any time its global maximum and
its global minimum in a prescribed compact region of
space. It was proven that with these particular settings
the statistically stationary régime exists and is very sim-
ilar to that arising in compact domains. In particular,
there exists a unique global minimizer located in a finite
spatial interval for all times and all other minimizers are
asymptotic to it in the limit t → −∞. The main idea
behind considering such type of forcing potential is to
ensure that the potential energy plays a dominant role
in comparison with the kinetic (elastic) term in the ac-
tion. This leads to effective compactification and allows
estimates on the velocities of fluid particles. As we al-
ready mentioned in section 5.1, these estimates are very
important and pave the way to the construction of the
whole theory of the statistically stationary régime.
Note finally that it was shown in [76] that for special
cases of forcing potentials F (x, t), the velocity of a min-
imizers can be arbitrarily large. More specifically, one
can construct pathological forcing potentials such that
minimizers are accelerated and reach infinite velocities.
Randomness is of course expected to prevent such a type
of non-generic blow-up.
6 Time-periodic forcing
This section is devoted to the study of the solutions
to the one-dimensional Burgers equation with time-
periodic forcing. In this case many of the objects we
have discussed above can be constructed almost explic-
itly: the global minimizer, the main shock etc. Also,
a mathematical analysis is then much simpler. For in-
stance, hyperbolicity of the global minimizer follows
immediately from first principles. Finally, the case of
time-periodic forcing is directly related to the Aubry-
Mather theory as we explain below.
6.1 Kicked Burgers turbulence
We shall be concerned here with the initial-value prob-
lem for the one-dimensional Burgers equation when the
force is concentrated in Dirac delta functions at discrete
times:
f(x, t) =
fj(x) δ(t − tj), (6.1)
where both the “impulses” fj(x) and the “kicking times”
tj are prescribed (deterministic or random). The kicking
times are ordered and form a finite or infinite sequence.
The impulses fj(x) are always taken spatially smooth,
i.e. acting only at large scales. The general scheme we
are presenting below holds for any sequence of impulses
fj(x) and kicking time. Later on we shall assume that
they define a time-periodic forcing. The precise meaning
we ascribe to the Burgers equation with such forcing is
that at time tj , the solution u(x, t) changes discontinu-
ously by the amount fj(x)
u(x, tj+) = u(x, tj−) + fj(x), (6.2)
while, between tj+ and t(j+1)− the solution evolves ac-
cording to the unforced Burgers equation.
We shall also make use of the formulation in terms of the
velocity potential ψ(x, t) and the force potentials Fj(x)
u(x, t) = −∂xψ(x, t), fj(x) = −
Fj(x). (6.3)
The velocity potential satisfies
∂tψ =
(∂xψ)
2 + ν∂xxψ +
Fj(x) δ(t − tj), (6.4)
ψ(x, t0) = ψ0(x), (6.5)
where ψ0(x) is the initial potential.
Using the variational principle we obtain the following
“minimum representation” for the potential in the limit
of vanishing viscosity which relates the solutions at any
two times t > t′ between which no force is applied:
ψ(x, t) = −min
(x− y)2
2(t− t′)
− ψ(y, t′)
. (6.6)
As before, when t′ is the initial time, the position y which
minimizes (6.6) is the Lagrangian coordinate associated
to the Eulerian coordinate x. The map y 7→ x is called
the Lagrangian map. By expanding the quadratic term it
is easily shown that the calculation of ψ(·, t) from ψ(·, t′)
is equivalent to a Legendre transformation. For details,
see [104,107].
We now turn to the forced case with impulses applied at
the kicking times tj . Let tJ(t) be the last such time before
t. Using (6.6) iteratively between kicks and changing
the potential ψ(y, tj+1) discontinuously by the amount
Fj+1(y) at times tj+1, we obtain
ψ(x, t) = − min
{yj}j0≤j≤J
[A({yj};x, t; j0)) − ψ0(yj0)] , (6.7)
A({yj};x, t; j0) ≡
(x − yJ)2
2(t− tJ)
(yj+1 − yj)2
2(tj+1 − tj)
−Fj+1(yj+1)
, (6.8)
where A(j0;x, t; {yj}) is called the action. We shall as-
sume that the force potential and the initial condition
are periodic in the space variable and the period is taken
to be unity. This assumption is very important for the
discussion below.
For a given initial condition at tj0 we next define a “min-
imizing sequence” associated to (x, t) as a sequence of
yj’s (j = j0, j0 + 1, . . . , J(t)) at which the right-hand
side of (6.7) achieves its minimum. Differentiating the
action (6.8) with respect to the yj ’s one gets necessary
conditions for such a sequence, which can be written as
a sequence of (Euler–Lagrange) maps
vj+1 = vj + fj(yj), (6.9)
yj+1 = yj + vj+1(tj+1 − tj)
= yj + (vj + fj(yj))(tj+1 − tj), (6.10)
where
yj − yj−1
tj − tj−1
. (6.11)
These equations must be supplemented by the initial
and final conditions:
vj0 = u0(yj0), (6.12)
x= yJ + vJ+1(t− tJ). (6.13)
It is easily seen that u(x, t) = vJ+1 = (x− yJ)/(t− tJ ).
Observe that the “particle velocity” vj is the velocity of
the fluid particle which arrives at yj at time tj and which,
of course, has remained unchanged since the last kick (in
Lagrangian coordinates). Equation (6.9) just expresses
that the particle velocity changes by fj(yj) at the the
kicking time tj .
Note that (6.9)-(6.10) define an area-preserving and (ex-
plicitly) invertible map.
As in the case of continuous-in-time forcing we can for-
mulate the Burgers equation in the half-infinite time in-
terval (−∞, t] without fully specifying the initial con-
dition u0(x) but only its (spatial) mean value 〈u〉 ≡
u0(x)dx.
The construction of the solution in a half-infinite time
interval is done by extending the concept of minimizing
sequence to the case of dynamics starting at t0 = −∞.
For a half-infinite sequence {yj} (j ≤ J), let us define the
action A({yj};x, t;−∞) by (6.8) with j0 = −∞. Such
a half-infinite sequence will be called a “minimizer” (or
“one-sided minimizer”) if it minimizes this action with
respect to any modification of a finite number of yj’s.
Specifically, for any other sequence {ŷj} which coincides
with {yj} except for finitely many j’s (i.e. ŷj = yj , j ≤
J − k, k ≥ 0), we require
A({ŷj};x, t; J − k) ≥ A({yj};x, t; J − k). (6.14)
Of course, the Euler–Lagrange relations (6.9)-(6.10) still
apply to such minimizers. Hence, if for a given x and t we
know u(x, t), we can recursively construct the minimizer
{yj} backwards in time by using the inverse of (6.9)-
(6.10) for all j < J and the final condition – now an
initial condition – (6.13) with vJ+1 = u(x, t). This is
well defined except where u(x, t) has a shock and thus
more than one value.
One way to construct minimizers is to take a sequence
of initial conditions at different times t0 → −∞. At each
such time some initial condition u0(x) is given with the
only constraint that it have the same prescribed value
for 〈u〉. Then, (finite) minimizing sequences extending
from t0 to t are constructed for these different initial
conditions. This sequence of minimizing sequences has
limiting points (sequences themselves) which are pre-
cisely minimizers (E et al. 1998). The uniqueness of such
minimizers, which would then imply the uniqueness of
a solution to the Burgers equation in the time interval
]−∞, t], can only be shown by using additional assump-
tions, for example for the case of random forcing or when
the forcing is time-periodic.
If 〈u〉 = 0, the sequence {yj} minimizes the action
A({yj};x, t;−∞) in a stronger sense. Consider any
sequence {ŷj} such that, for some integer P we have
ŷj = yj + P , j ≤ J − k, k ≥ 0 and which differs ar-
bitrarily from {yj} for j > J − k. (In other words, in
a sufficiently remote past the hatted sequence is just
shifted by some integer multiple of the spatial period.)
We then have
A({ŷj};x, t;−∞) ≥ A({yj}, x, t;−∞). (6.15)
Indeed, for 〈u〉 = 0, the velocity potential for any initial
condition is itself periodic. In this case a particle can be
considered as moving on the circle S1 and its trajectory
is a curve on the space-time cylinder. The yj ’s are now
defined modulo 1 and can be coded on a representative
0 ≤ yj < 1. The Euler–Lagrange map (6.9)-(6.10) is still
valid provided (6.10) is defined modulo 1.
The condition of minimality implies now that yj and
yj+1 are connected by the shortest possible straight seg-
ment. It follows that |vj+1| = ρ(yj , yj+1)/(tj+1 − tj),
where ρ is the distance on the circle between the points
yj, yj+1, namely ρ(a, b) ≡ min{|a−b|, 1−|a−b|}. Hence,
the actionA can be rewritten in terms of cyclic variables:
A({yj};x, t;−∞) =
ρ2(x, yJ )
2(t− tJ)
ρ2(yj+1, yj)
2(tj+1 − tj)
− Fj+1(yj+1)
. (6.16)
The concept of “global minimizers” can be defined in
a usual way. Namely, global minimizers correspond to
one-sided minimizers that can be continued to a bilat-
eral sequence {yj ,−∞ < j < +∞} while keeping the
minimizing property. Such global minimizers correspond
to trajectories of fluid particles that, from t = −∞ to
t = +∞, have never been absorbed in a shock. As before
we define a “main shock” as a shock which has always
existed in the past.
From now on we shall consider exclusively the case where
the kicking is periodic in both space and time. Specifi-
cally, we assume that the force in the Burgers equation
is given by
f(x, t) = g(x)
δ(t− jT ), (6.17)
g(x) ≡− d
G(x), (6.18)
where G(x), the kicking potential, is a deterministic
function of x which is periodic and sufficiently smooth
(e.g. analytic) and where T is the kicking period. The
initial potential ψinit(x) is also assumed smooth and pe-
riodic. This implies that the initial velocity integrates to
zero over the period. The case where this assumption is
relaxed will be considered later in connection with the
Aubry–Mather theory.
The numerical experiments of [9] reported here have
Fig. 28. Snapshots of the velocity for the unique time-peri-
odic solution corresponding to the kicking force g(x) shown
in the upper inset; the various graphs correspond to six out-
put times equally spaced during one period. The origin of
time is taken at a kick. Notice that during each period, two
new shocks are born and two mergers occur. (From [9].)
been made with the kicking potential
G(x) =
sin 3x+ cosx, (6.19)
and a kicking period T =1. Other experiments were done
with (i) G(x) = − cosx and (ii) G(x) = (1/2) cos(2x) −
cosx. The former potential produces a single shock and
no preshock. As a consequence it displays no −7/2 law
in the PDF of gradients. The latter potential gives es-
sentially the same results as reported hereafter but has
an additional symmetry. To avoid non-generic behaviors
that could result from this symmetry, it was chosen to
focus on the forcing potential given by (6.19).
The number of collocation points chosen for such simu-
lations was mostly Nx = 2
17 ≈ 1.31 × 105, with a few
simulations done at Nx = 2
20 (for the study of the re-
laxation to the periodic régime presented below). Since
the numerical method allows going directly to the de-
sired output time (from the nearest kicking time) there
is no need to specify a numerical time step. However, in
order to perform temporal averages, e.g. when calculat-
ing PDF’s or structure functions, without missing the
most relevant events (which can be sharply localized in
time) sufficiently frequent temporal sampling is needed.
The total number of output times Nt ≈ 1000, is thus
chosen such that the increment between successive out-
put times is roughly the two-thirds power of the mesh
(this is related to the cubic structure of preshocks, see
section 2.3).
Figure 28 shows snapshots of the time-periodic solution
at various instants. It is seen that shocks are always
present (at least two) and that at each period two new
shocks are born at t⋆1 ≈ 0.39 and t⋆2 ≈ 0.67. There is one
main shock which remains near x = π and which collides
space x
Main Shock 
Fig. 29. Evolution of shock positions during one period. The
beginnings of lines correspond to births of shocks (preshocks)
at times t⋆1 and t⋆2; shock mergers take place at times tc1
and tc2. The “main shock”, which survives for all time, is
shown with a thicker line.
0 2 4 6 8 10 12 14
number of kicks
sin(x) 
sin(2x)
sin(3x)
slope −0.74 
Fig. 30. Exponential relaxation to a time-periodic solution
for three different initial velocity data as labeled. The hori-
zontal axis gives the time elapsed since t = 0. (From [9].)
with the newborn shocks at tc1 ≈ 0.44 and tc2 ≈ 0.86.
Figure 29 shows the evolution of the positions of shocks
during one period.
It was found that, for all initial conditions u0(x) used, the
solution u(x, t) relaxes exponentially in time to a unique
function u∞(x, t) of period 1 in time. Figure 30 shows the
variation of
|u(x, n−)−u∞(x, 1−)| dx/(2π) for three
different initial conditions as a function of the discrete
time n.
The phenomenon of exponential convergence to a unique
space- and time-periodic solution is something quite gen-
eral: whenever the kicking potentialG(x) is periodic and
analytic and the initial velocity potential is periodic (so
that the mean velocity 〈u〉 =0 at all times), there is ex-
ponential convergence to a unique piecewise analytic so-
lution. This can be proved rigorously (see Appendix to
[9]) in the case when the functions G(x) have a unique
point of maximum with a non-vanishing second deriva-
tive (Morse generic functions). Here, we just explain the
main ideas of the proof and give some additional prop-
erties of the unique solution.
One very elementary property of solutions is that, for
any initial condition of zero mean value, the solution
after at least one kick satisfies
|u(x, t)| ≤ (1/2) + max
|dG(x)/dx|. (6.20)
Indeed, at a time t = n− just before any kick we have
x = y+u(x, n−) where y is the position just after the pre-
vious kick of the fluid particle which goes to x at time n−.
It follows from the spatial periodicity of the velocity po-
tential that the location y which minimizes the action is
within less than half a period from x. Thus, |u(x, n−)| ≤
1/2. The additional maxx |dG(x)/dx| term comes from
the maximum change in velocity from one kick. Hence
the solution is bounded. Note that if the spatial and tem-
poral periods are L and T , respectively, the bound on
the velocity becomes L/(2T ) + maxx |dG(x)/dx|.
The convergence at large times to a unique solution can
be understood in terms of the two-dimensional conserva-
tive (area-preserving) dynamical system defined by the
Euler–Lagrange map (6.9)-(6.10). By construction, we
have u(x, 1+) = û(x) − dG(x)/dx, where û(x) is the so-
lution of the unforced Burgers equation at time t = 1−
from the initial condition u(x) at time t = 0+. The map
u 7→ û(x) + g(x), where g(x) ≡ −dG(x)/dx, here de-
notedBg, solves the kicked Burgers equation over a time
interval one. The problem is to show that the iterates
Bng u0 converge as n→ ∞ to a unique solution.
If it were not for the shocks it would suffice to consider
the two-dimensional Euler–Lagrange map. Note that,
for the case of periodic kicking, this map has an obvi-
ous fixed point P , namely (x = xc, v = 0), where xc is
the unique point maximizing the kicking potential. It is
easily checked that this fixed point is an unstable (hy-
perbolic) saddle point of the Euler–Lagrange map with
two eigenvalues λ = 1 + c +
c2 + 2c and 1/λ, where
c = −∂2xxG(xc)/2.
Like for any two-dimensional map with a hyperbolic
fixed point, there are two curves globally invariant by
the map which intersect at the fixed point. The first is
the stable manifold Γ(s), i.e. the set of points which con-
verge to the fixed point under indefinite iteration of the
map; the second is the unstable manifold Γ(u), i.e. the
set of points which converge to the fixed point under in-
definite iteration of the inverse map, as illustrated in fig-
ure 31(a). Any curve which intersects the stable manifold
transversally (at the intersection point, the two curves
xx xl rc
Fig. 31. (a) Sketch of a hyperbolic fixed point P with stable
(Γ(s)) and unstable (Γ(u)) manifolds. The dashed line gives
the orbit of successive iterates of a point near the stable
manifold. (b) Unstable manifold Γ(u) on the (x, v)-cylinder
(the x-coordinate is defined modulo 1) which passes through
the fixed point P = (xc, 0). The bold line is the graph of
u∞(x, 1−). The main shock is located at xl = xr. Another
shock at x1 corresponds to a local zig-zag of Γ
(u) between A
and B.
are not tangent to each other) will, after repeated appli-
cations of the map, be pushed exponentially against the
unstable manifold at a rate determined by the eigenvalue
1/λ. In the language of Burgers dynamics, the curve in
the (x, v) plane defined by an initial condition u0(x) will
be mapped after time n into a curve very close to the
unstable manifold. In fact, for the case studied numeri-
cally, 1/λ ≈ 0.18 is within one percent of the value mea-
sured from the exponential part of the graph shown in
figure 30. Note that if the initial condition u0(x) contains
the fixed point, the convergence rate becomes (1/λ)
(even higher powers of 1/λ are possible if the initial con-
dition is tangent to the unstable manifold).
The fixed point P is actually a very simple global min-
imizer: (yj = xc, vj = 0) for all positive and negative
j’s. It follows indeed by inspection of (6.16) that any
deviation from this minimizer can only increase the ac-
tion; actually, this trajectory minimizes both the kinetic
and the potential part of the action. Note that the cor-
z  (0)l rz  (0)
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����������������������������������������������
����������������������������������������������
����������������������������������������������
����������������������������������������������
z  (0)1z c
Fig. 32. Minimizers (trajectories of fluid particles) on the
(x, t)-cylinder. Time starts at −∞. Shock locations at t = 0−
are characterized by the presence of two minimizers (an in-
stance is at x1). The main shock is at xl = xr. The fat line
x = xc is the global minimizer.
responding fluid particle is at rest forever and will never
be captured by a shock (it is actually the only particle
with this property). It is easy to see that any minimizer
is attracted exponentially to such a global minimizer as
t→ −∞. Thus, any point (yj , vj) on a minimizer belongs
to the unstable manifold Γ(u) and, hence, any regular
part of the graph of the limiting solution u∞(x) belongs
to the unstable manifold Γ(u). This unstable manifold is
analytic but can be quite complicated. It can have sev-
eral branches for a given x (see figure 31(b)) and does
not by itself define a single-valued function u∞(x). The
solution has shocks and is only piecewise analytic. Con-
sideration of the minimizers is required to find the po-
sition of the shocks in the limiting solution: two points
with the same x corresponding to a shock, such as A and
B on figure 31(b) should have the same action.
Finally, we give the geometric construction of the main
shock, the only shock which exists for an infinite time.
Since the eigenvalue λ is positive, locally, minimizers
which start to the right of xc approach the global min-
imizer from the right, and those which start to the left
approach it from the left. Take the rightmost and left-
most points xr and xl on the periodicity interval such
that the corresponding minimizers approach the global
minimizer from the right and left respectively (see fig-
ure 32). These points are actually identical since there
cannot be any gap between them that would have min-
imizers approaching the global minimizer neither from
the right nor the left. The solution u∞(x) has then its
main shock at xl = xr.
6.2 Connections with Aubry–Mather theory
In the previous subsection, the study of the solutions
to the periodically kicked Burgers equation was limited
to initial conditions with a vanishing spatial average
b. With a non-vanishing mean velocity b, which in the
forced case cannot be eliminated by a Galilean invari-
ance, many of the properties of the solutions described
above are still valid. However the action now depends on
b. Global minimizers {y(g)j , j ∈ Z} exist in this case as
well. However generically they are not unique and do not
correspond to fixed points of the Euler–Lagrange map
(6.9)-(6.10). A global minimizer now minimizes the ac-
A∞({yk}) =A({yk}; +∞;−∞)
(yk+1−yk−b)2−G(yk+1)
. (6.21)
This action is exactly the potential energy associated
to an infinite chain of atoms linked by elastic springs
and embedded in a periodic potential, problem known
as the Frenkel–Kontorova model [52]. The parameter b
represents the equilibrium length l of the springs and
the spatial period L of the external potential (see fig-
ure 33) is equal to 1. A global minimizer of (6.21) rep-
Fig. 33. Sketch of the Frenkel–Kontorova model for the equi-
librium states of an atom chain in a periodic potential.
resents an equilibrium configuration of this system. The
properties of this equilibrium, or ground states are de-
termined by the competition between two tendencies: on
the one hand the atoms tend to stabilize at those loca-
tions where the potential is minimum; on the other hand,
the springs tend to maintain them at a fixed distance of
each other. When b = 0 this competition disappears and
the equilibrium is given by yk = xc, where xc is the loca-
tion at which G attains its global minimum. For b 6= 0,
the situation is more delicate and the structure of the
ground states involves, as we shall now see, a problem
of commensurate-incommensurate transition. The prop-
erties of ground states were studied in great details by
Aubry [3] and Mather [86].
The relations between the Burgers equation with a
time-periodic forcing and Aubry–Mather theory were
discussed for the first time in [70] and in [38]. The the-
ory was further developed in [36,106]. For integer values
of b, the global minimizer is trivially associated to the
fixed point (x, v) = (xc, b) of the Euler–Lagrange map
(6.9)-(6.10), which corresponds to a fluid trajectory lo-
cated at integer times at x = xc and which moves on
distance of b spatial periods during one temporal period.
A similar argument implies that it is enough to study
values of b in the interval [0, 1).To each global minimizer
{y(g)j , j ∈ Z} is associated a rotation number defined as
ρ ≡ lim
j+1 − y
, (6.22)
which represents the time-average velocity of the mini-
mizer. For a fixed value of the spatial average b of the
velocity, all global minimizers associated to the solution
of the Burgers equation have the same rotation number
ρ. Indeed, as the dynamics is restricted to a compact do-
main of the configuration space (in our case T), two min-
imizers with different rotation numbers necessarily cross
each other; this is an obvious obstruction to the action
minimization property. In the case of rational rotation
numbers the global minimizers correspond to periodic
orbits of the dynamical system defined by the Euler–
Lagrange map. An important feature is that for rational
ρ, the rotation number does not change when varying
b over a certain closed interval [bmin, bmax], called the
mode-locking interval. On the contrary, irrational ρ cor-
respond to a unique value of the parameter b. Such “ir-
rational” values of b form a Cantor set of zero Lebesgue
measure. In particular, the graph of ρ as a function of
the parameter b is a “Devil staircase” (see figure 34).
0 0.2 0.4 0.6 0.8 1
Fig. 34. Rotation number ρ as a function of the spatial mean
of the velocity b for the standard map.
When ρ is rational (ρ = p/q in irreducible form), global
minimizers correspond to a periodic orbit of period q.
It is easy to see that such an orbit generates q different
but closely related global minimizers. Of course each of
these global minimizer is the image of another one by the
Euler–Lagrange map and is mapped back to itself after
q iterations. This procedure generates a periodic orbit,
which turns out to be hyperbolic one. Hence, each of
the q global minimizers has a one-dimensional unstable
manifold associated to it. The solution to the Burgers
equation is formed by branches of these various mani-
folds with jumps between them defining q global shocks.
The picture is very different for values of b corresponding
to irrational rotation numbers. Consider velocities and
positions of all global minimizers at a fixed moment of
time, say t = 0. They form a subset G of the phase space
T × R. Then two cases have to be distinguished:
• The set G forms a closed invariant curve for the Euler–
Lagrange map. This invariant curve has a one-to-one
projection onto the base T and dynamics on the curve
is conjugated to a rigid rotation by angle ρ. The lim-
iting solution of the Burgers equation is given by the
invariant curve and does not contain any shocks.
• The set G forms an invariant Cantor set and the limit-
ing solution of the Burgers equation contains an infi-
nite number of shocks, none of which is a main shock.
The Kolmogorov [80], Arnold [2] and Moser [93] the-
ory (frequently referred to as KAM) describes invari-
ant curves (or invariant tori) for small analytic per-
turbations of integrable Hamiltonian systems, and thus
the various types of dynamical trajectories. The KAM
theory ensures that for sufficiently small perturbations,
most of the invariant curves associated to Diophantine
irrational rotation numbers are stable with respect to
small analytic perturbations of the system. Diophantine
irrational numbers possess fast converging approxima-
tions by rational numbers (in a suitable technical sense).
However, these invariant curves may disappear from the
perturbed system when an interaction corresponding to
a non-integrable perturbation gets sufficiently strong.
Aubry–Mather theory provides another variational de-
scription for the KAM invariant curves. But even more
importantly, it describes the invariant Cantor sets that
appear instead of invariant curves in the case of strong
nonlinear interactions. We have mentioned already that
these Cantor sets correspond to global minimizers. Thus
Aubry–Mather theory provides information about the
global minimizers and, hence, allows one to study in such
a situation the properties of limiting entropic solutions
and, in particular, the structure of shocks.
A numerical study of the Burgers equation in the inviscid
limit, with periodic forcing and a non-vanishing spatial
average of the velocity, reveals the appearance of shock
accumulations. Such events occur for the values of the
mean velocity b near the end-points of the mode-locking
intervals, corresponding to rational rotation numbers.
The shock accumulation phenomenon is due to the fact
that the end-points bmin, bmax of the mode-locking in-
tervals can be approximated by convergent sequences of
“irrational” values of the parameter b. This implies ac-
cumulation of shocks, since for irrational rotation num-
bers the number of shocks is infinite.
The limiting solution u∞(x, t) is completely determined
by the function û(x) defined in the previous subsection.
The function û(x) corresponds to a stroboscopic section
of u∞ right after each impulse. The regular parts of û
are made of single-valued functions related to the unsta-
ble manifolds. The shocks correspond to jumps, either
between different branches of the same manifold (sec-
ondary shocks), or between the manifolds associated to
different global minimizers (main shocks).
When the rotation number is rational (ρ = p/q), there
are q global minimizers. The positions of the q main
shocks of û are determined by a requirement that the
area defined by the graph of the solution is equal to the
conserved quantity b. The latter constraint shows that
the values of b compatible with the rotation number p/q
belong to an interval [bmin, bmax] bounded by the mini-
mum and maximum areas defined by the unstable man-
ifolds, as illustrated in figure 35. The detailed shape of
Fig. 35. Sketch of the unstable manifolds of the two global
minimizers associated to the rotation number ρ = 1/2. The
values bmin and bmax given by this configurations are repre-
sented as grey areas.
the manifolds can actually not be sketched on a figure.
Generically the unstable manifold of a global minimizer
corresponding to a particular point of the basic periodic
orbit of period q intersects transversally with the stable
manifold of another minimizer corresponding to another
point of the periodic orbit. Such an intersection leads to
formation of a heteroclinic tangle, a notion which can be
traced back to the work of Poincaré. The heteroclinic in-
tersection results in the formation of an infinite number
of zig-zags of the unstable manifolds. These zig-zags are
accumulating along the stable manifold and come arbi-
trary close to the corresponding point of the periodic or-
bit. The zig-zags contract exponentially in one direction
(along the stable manifold) and are stretched exponen-
tially in the other direction. It is easy to see that the
accumulation of zig-zags generates an infinite number of
“potential” shocks of smaller and smaller size which also
accumulate near the periodic orbit. When the param-
eter b is located well inside the mode-locking interval,
the position of the main shock cuts off the accumulated
shocks of small size so that the total number of shocks
is of the order of unity. However, when b gets closer and
closer to bmax or bmin, the main shocks move closer to
the periodic points and a larger number of the small ac-
cumulating shocks appears in the solution. This mecha-
nism leads to an infinite number of shocks in the solution
when b is equal to bmin or bmax (see figure 36(a)). Both
Fig. 36. (a) Accumulations of shocks occurring for b = bmin or
b = bmax, due to the presence of an infinite number of loops
of the unstable manifold in the homocline or heterocline
tangle. (b) Shock accumulation at the fixed point (0, 0) of
the standard map. Here, λ = 0.1 and b = 0.15915. The latter
value is close to the upper bound of the interval associated
to ρ = 0. The upper inset is a zoom near (0, 0), illustrating
the accumulation of shocks.
the distances between two consecutive shocks and the
sizes of the shocks decrease exponentially fast with the
number of shocks; the rate is given by the stable eigen-
value associated to the hyperbolic periodic orbit. It is
interesting to mention that when b = bmin or b = bmax
the main shocks merge with the periodic orbit associ-
ated to the global minimizers. Hence, for the end-points
of the mode-locking interval the main shocks disappear.
To illustrate numerically the change in behavior of the
solution to the Burgers equation when the mean velocity
b changes, we focus here on the simple periodic kicking
potential G(x) = (λ/2π) cos(2πx) where λ is a free pa-
rameter. The associated Euler–Lagrangemap then reads
T : (y, v) 7→(y+v+λ sin(2πy), v+λ sin(2πy)). (6.23)
This transformation is usually called the standard map
(or Chirikov–Taylormap). It is one of the simplest model
for studying the presence of chaos in Hamiltonian dy-
Fig. 37. General aspect in position-velocity phase space of
the dynamical system defined by the standard map (6.23)
for two different values of the parameter (a) λ = 0.1 and
(b) λ = 0.3. The corresponding time-periodic solutions to
the kicked Burgers equation are represented as bold lines in
both cases. The results are presented for the spatial mean
velocities b = 0, b = 0.3 and b = 0.5.
namical systems and in particular particularly to study
the KAM theory.
Figure 36(b) illustrates the accumulation of shocks due
to the homoclinic or heteroclinic tangling for the first
transition (starting from b = 0). This transition cor-
responds to a rotation number of the global minimizer
changing value from ρ = 0 to ρ > 0. When b is increased
and gets close to the critical value, shocks accumulate
on the left-hand side of the global minimizer located at
(y, v) = (0, 0).
Other numerical experiments were performed in order
to observe the destruction of invariant curves and the
accumulation of shocks on Cantor sets for irrational ro-
tation numbers. It is of course impossible numerically to
set the rotation number to an irrational value. Indeed,
the values of b for which ρ is irrational are in a Cantor
set. It is however possible to be very close to irrational
rotation numbers. Figure 37 illustrates the changes in
the behavior of the solutions to the periodically kicked
Burgers equation when varying the parameter λ. The
time-asymptotic solutions associated to various values
of the mean velocity b are shown for λ = 0.1 and λ = 0.3.
For the latter value, all KAM invariant curves have al-
ready disappeared. For b = 0 and for all values of λ the
global minimizer trivially corresponds to the fixed point
(0, 0) with a vanishing rotation number. For b = 0.5
there are two global minimizers associated to the ratio-
nal rotation number ρ = 1/2. For λ = 0.1 and b = 0.3
the rotation number is much closer to an irrational than
in previous cases. The solution is then very close to the
invariant curve associated to this value. Note that the
main shock is actually located close to x ≈ 0.85. It is so
small that it can hardly be seen. When λ = 0.3 the value
b = 0.3 of the mean velocity no more corresponds to a
rotation number close to an irrational value; it is now
in the mode-locking interval associated to ρ = 1/3. This
change in the rotation number reflects the dependence of
the mode-locking intervals [bmin, bmax] on the parameter
λ. The interval of values of b associated to ρ = 0 is rep-
resented as a function of λ in figure 38. Such a structure
is frequently called an Arnold tongue (see, e.g., [72]).
−1.5 −1 −0.5 0 0.5 1 1.5
Fig. 38. Evolution as a function of the parameter λ of the
mode-locking interval [bmin, bmax] associated to the rotation
number ρ = 0. Such a graph is frequently referred to as an
Arnold tongue.
Finally, we discuss the structure of shocks in the case
when the global minimizers form a Cantor set. There are
then infinitely many gaps with no global minimizers. It
is known in this case that all the gaps can be split into
the finite number of images of the main gaps. For the
standard map there is only one main gap. Its end-points
(x1, v1) and (x2, v2) belong to the Cantor set associated
to the global minimizers. All other gaps can be obtained
by iterating this main gap with the Euler-Lagrange map
(Standard map) for both positive and negative times:
(x1i , v
i ) = T i(x1, v1), (x2i , v2i ) = T i(x2, v2), i ∈ Z. One
can show that the length of the ith gap tends to zero as
i→ ±∞. Since global minimizers are hyperbolic trajec-
tories one can connect the end-points of the main gap
by two smooth curves: the stable manifold Γ(s) and the
unstable manifold Γ(u). As i→ ∞ the iterates of the sta-
ble manifold T iΓ(s) tend to a straight segment connect-
ing the i-th gap with end-points at (x1i , v
i ) and (x
i , v
The same is true for iterates of the unstable manifold
T iΓ(u) in the limit i → −∞. On the contrary, negative
iterates of the stable manifold and positive of the unsta-
ble one form exponentially long curves connecting corre-
sponding gaps. As usual we are interested in the iterates
of the unstable manifold since they appear in the time-
periodic solution of the Burgers equation. Such a solu-
tion is formed by the iterates of the unstable manifold
connecting all the gaps. Note that in the case of large
negative i, the unstable manifold is close to a straight
segment; hence there are no shocks located inside the
corresponding gaps. Conversely, for large positive i, the
unstable manifold is exponentially long and possesses
large zig-zags. Hence, the solution to the Burgers equa-
tion has one or several shocks inside such gaps. Since
there are no shocks for gaps with large enough negative
i, it follows that all the shocks have a finite age. In other
words, the time-periodic solution has no main shocks.
At the moment it was not possible to study numeri-
cally the strange behavior of the solutions to the Burgers
equation corresponding to global minimizers living on
Cantor-like sets. Looking for such cases requires a very
high spatial resolution in order to minimize the numeri-
cal error in the approximation of the solution. Moreover,
a large number of values for the parameters b and λ has to
be investigated in order to observe such a phenomenon.
This would require heavy computer ressources. How-
ever, many other aspects of the Aubry–Mather theory
for Hamiltonian systems can be studied numerically us-
ing the Burgers equation with periodic kicks. For in-
stance it could be very useful for analyzing the higher
dimensional versions.
7 Velocity statistics in randomly forced Burgers
turbulence
The universality of small-scale properties in fully devel-
oped Navier–Stokes turbulence has frequently been in-
vestigated, assuming that a steady state is maintained
by an external large-scale forcing. It is generally conjec-
tured that the velocity increments have universal sta-
tistical properties with respect to such a force. Under-
standing this issue in simpler models of turbulence has
motivated much work for over ten years. A toy model
which has been extensively studied is the passive trans-
port of a scalar field by random flows (see, e.g., [46]).
Tools borrowed from statistical physics and field theory
were used to describe and explain the anomalous scaling
laws observed in the scalar spatial distribution. It was
shown that the scale invariance symmetry is broken by
geometrical constraints on tracer configurations that are
statistically conserved by the dynamics. Universality of
the intermittent scaling exponents with respect to the
forcing was proven for the case where energy is injected
at large scales [31,57,103,14].
Issues of universality for the nonlinear Burgers turbu-
lence model has also been very much on the focus. The
possibility to solve exactly a hydrodynamical problem
displaying the same kind of quadratic nonlinearity as
Navier–Stokes turbulence constitutes of course the cen-
tral motivation. Three independent approaches were
published almost simultaneously in 1995 and were at
the origin of the growing interest in Burgers turbulence.
First, an analogy was made in [22] between forced Burg-
ers turbulence and the problem of a directed polymer in
a random medium. This analogy was used to show that
the shocks appearing in the solution lead to anomalous
scaling laws for the structure functions. The strong
intermittency could be related to the replica-symmetry-
breaking nature of the disordered system associated
to Burgers turbulence. This approach is discussed in
subsection 7.1. Second, ideas using operator product
expansions borrowed from quantum field theory were
proposed in [99]. The goal was to close in the inertial
range the equations governing the correlations of the ve-
locity field in one dimension. This treatment of the dis-
sipative anomaly is described in subsection 7.2. It yields
a prediction for the probability density function (PDF)
of velocity increments and gradients and in particular
to a power-law behavior for the PDF of ∂xv at large
negative values [99]. However, the value of the exponent
of this algebraic tail has been a matter of controversy.
An overview of the various works related to this issue is
given in subsection 7.3. Finally, the turbulent model of
the one-dimensional Burgers equation with a self-similar
forcing was proposed in [30] as one of the simplest non-
linear hydrodynamical problem displaying multiscaling
of the velocity structure function. As stressed in subsec-
tion 7.4 this problem is easily tractable numerically and
some of the numerical observations can be confirmed by
a one-loop renormalization group expansion.
In what follows we consider the solutions to the Burg-
ers equation with a homogeneous Gaussian random forc-
ing that is delta-correlated in time. Namely, the spatio-
temporal correlation of the forcing potential is taken to
〈F (~x, t)F (~x′, t′)〉 = B(~x − ~x′) δ(t− t′) . (7.1)
The function B contains information on the spatial
structure of the forcing. It can be either smooth (i.e.
concentrated at large spatial scales) or asymptoti-
cally self-similar (i.e. behaving as a power law at small
separations). In the former case the solution reaches
exponentially fast a statistically stationary régime in
any space dimension. The construction of the solution
in this régime in terms of global minimizer and main
shock is described in detail in section 5. When B does
not decrease sufficiently fast at small separations (e.g.
B(r) ∼ r2h with h < 1 as r → 0 in one dimension), there
is no rigorous proof of the existence of a statistically sta-
tionary régime. However we assume in the sequel that
such a stationary régime exists in order to perform a
statistical analysis of the solutions to Burgers equation.
7.1 Shocks and bifractality – a replica variational ap-
proach
The replica solution for Burgers turbulence proposed
in [22] is based on its analogy with the problem of a di-
rected polymer in a random medium. As already stated
in the Introduction, the viscous Burgers equation forced
by the potential F is equivalent to finding the partition
function Z of an elastic string in the quenched spatio-
temporal disorder V (~x, t) = F (~x, t)/2ν (remember that
t has to be interpreted as the space direction in which
the polymer is oriented). This relation is obtained by ap-
plying to the velocity potential Ψ the Hopf–Cole trans-
formation Z(~x, t)=exp(Ψ(~x, t)/2ν). The solution of the
problem can be written in terms of the path integral
Z(~x, t) =
~γ(t)=~x
exp(−H(~γ)) d[~γ(·)] ,
with H(~γ) = 1
∥~̇γ(s)
+ F (~γ(s), s)
ds. (7.2)
In the analogy between Burgers turbulence and directed
polymers, the polymer temperature is assumed to be
unity and its elastic modulus is 1/(2ν). The strength
of the potential fluctuations applied to the polymer de-
pends on the viscosity and is ∝ ε1/2Lf/(2ν) (where ε is
the energy injection rate and Lf is the spatial scale of
forcing). In order to calculate the various moments of the
velocity field ~v = −∇Ψ, one needs to average the loga-
rithm of the partition function Z, a celebrated problem
in disordered systems.
Bouchaud, Mézard and Parisi proposed in [22] the use
of a replica trick in order to estimate the average free
energy 〈lnZ〉. The first step is to write the zero-replica
limit lnZ = limn→0 (Zn − 1)/n. Then, the moments
〈Zn〉 are used to generate an effective attraction between
replicas: they are written as the partition functions of
the disorder-averaged Hamiltonian Hn(~γ1, . . . , ~γn) asso-
ciated to n replicas of the same system [90]
∥~̇γi(s)
B(~γi(s)−~γj(s))
,(7.3)
where B denotes the spatial part of the forcing potential
correlation. The next step is to study this problem by a
variational approach. The Hamiltonian Hn is replaced
by an effective Gaussian quadratic Hamiltonian that can
be written as
Heff =
~γi(τ)Gij(τ−τ ′)~γj(τ ′)dτdτ ′. (7.4)
The kernel Gij is then chosen in such a way that it
minimizes the free energy. It is shown in [22] that the
optimal Gaussian Hamiltonian is the solution of a sys-
tem of equations that can be solved following the ansatz
proposed in [89]. When d > 2 this approach singles
out two régimes depending on the Reynolds number
Re = ε1/3L
f /ν. These régimes are separated by the
critical value Rec = [2(1−2/d)1−d/2]1/3. When Re < Rec
the optimal solution is of the form Gij = G0 δij +G1 and
obeys the replica symmetry. In finite-size systems it cor-
responds to a linear velocity profile. When Re > Rec
the correct solution is given by the one-step replica-
symmetry-breaking scheme (see [89]). The off-diagonal
elements of Gij are then parameterized with two func-
tions depending on whether the indices i and j belong to
the same block or to different blocks. Qualitatively, the
one-step replica-symmetry-breaking approach amounts
to the assumption that the instantaneous velocity po-
tential can be written as a weighted sum of Gaussians,
leading to an approximation of the velocity field as
~v(~x, t) ≃
α(~x− ~rα) e
−Re (fα+‖~x−~rα‖
2/2L2
−Re (fα+‖~x−~rα‖2/2L2f )
, (7.5)
where the fα’s are independent variables with a Poisson
distribution of density exp(−f). The ~rα are uniformly
and independently distributed in space. In (7.5) the sum
over α is running from 1 to a large-enough integer M .
The typical shape of the approximation of the veloc-
ity field given by (7.5) is represented in figure 39(a) in
the two-dimensional case. In the limit of large Reynolds
numbers the random velocity field given by (7.5) typi-
cally contains cells of width ∝ Lf . The width of a shock
separating two cells is of the order of Lf/Re.
The replica approximation (7.5) leads to an estimate of
the PDF p(∆v, r) of the longitudinal velocity increment
∆v = (~v(~x+ r ~e, t) − ~v(~x, t)) · ~e, where ~e is an arbitrary
unitary vector. When Re ≫ 1 and r ≪ Lf this approxi-
mation takes the particularly simple asymptotic form
p(∆v, r) ≈ δ
∆v − Uf
, (7.6)
where Uf = Re ν/Lf is the typical velocity associated
to the scale Lf and g is a scaling function that is deter-
mined explicitly in [22]. This approximation is in agree-
ment with the following qualitative picture. With a prob-
ability almost equal to one, the two points ~x and ~x+ r ~e
lie in the same cell; the velocity increment is then given
by the typical velocity gradient which, according to the
Fig. 39. (a) Typical shape of the velocity field given by the
replica approximation in dimension d = 2 obtained from
(7.5) for Re = 103. The contour lines represent the velocity
modulus. Note the cell structure of the domain. (b) Scaling
exponents of the pth order structure function.
approximation (7.5), is order Uf/Lf . With a probabil-
ity r/Lf the two points are sitting on different sides of
a shock separating two such cells and the associated ve-
locity difference is of the order of Uf .
The structure functions of the velocity field given by
the various moments of ∆v can be straightforwardly
estimated from the approximation (7.6). Their scaling
behavior 〈∆vp〉 ∼ rζp at small separations r display
a bifractal behavior as sketched in figure 39(b). When
p < 1, the first term on the right-hand side of (7.6) domi-
nates and 〈∆vp〉 ∝ Upf (r/Lf )p. For p > 1 the shock con-
tribution is dominating the small-r behavior and thus
〈∆vp〉 ∝ Upf (r/Lf ).
This approach, which makes use of replica tricks, is as we
have seen able to catch the leading scaling behavior of
velocity structure functions in any dimension. It is based
on approximations of the velocity field by the superposi-
tion (7.5) of Gaussian velocity potentials. A first advan-
tage of this method is that it catches the generic aspect
of the solution including the hierarchy of high-order sin-
gularities appearing in the solution when Re → ∞ which
was examined in section 2.3. This method also gives
predictions regarding the dependence on Re of the sta-
tistical properties of the solution. However, as stressed
in [22], the validity of this approximation is expected to
hold only in the limit of infinite space dimension d. In
particular, it is known that for d ≤ 2 a full continuous
replica-symmetry-breaking scheme is needed [89]. Nev-
ertheless, as we have seen, there is enough evidence that
this approach describes very well the qualitative aspects
of the solution.
7.2 Dissipative anomaly and operator product expan-
The replica-trick approach described in the previous
subsection cannot reproduce one of the main statis-
tical features of the solution, namely the tails of the
velocity increments PDF p(∆v, r). Indeed the predic-
tion (7.6) based on a variational approximation of the
velocity field implies that p identically vanishes when
∆v > Uf (r/Lf ). In order to study the quantitative be-
havior of the PDF p(∆v, r) in the inviscid limit ν → 0
(or equivalently Re → ∞), Polyakov [99] proposed to
use an operator product expansion. This approach leads
to an explicit expression for p(∆v, r) and predicts a
super-exponential tail at large positive values and a
power-law behavior for negative ones. Such predictions
have immediate implications for the asymptotics of the
PDF p(ξ) of the velocity gradient ξ = ∂xv. The work
of Polyakov was the starting point of a controversy on
the value of the exponent of the left tail of p(ξ). Before
returning to this issue in the next subsection, we give
in the sequel a quick overview of the original work by
Polyakov.
We henceforth focus on the one-dimensional solutions
to the Burgers equation with Gaussian forcing whose
autocorrelation is given by (7.1). Following [99] (see also
[19,20]) we introduce the characteristic function of the
n-point velocity distribution
Zn(λj , xj ; t) ≡
e λ1 v(x1,t)+···+λn v(xn,t)
. (7.7)
For a finite value of the viscosity ν, it is easily seen that
this quantity is a solution to a Fokker–Planck (master)
equation obtained by differentiating Zn with respect to
t and using the Burgers equation and the fact that the
forcing is Gaussian and δ-correlated in time. This leads
b(xi − xj)λi λj Zn + D(n)ν , (7.8)
where b ≡ (d2B)/(dr2) denotes the spatial part of the
correlation of the forcing applied to the velocity field.
D(n)ν denotes the contribution of the dissipative term and
reads
D(n)ν ≡ ν
v(xj , t)
λj v(xj ,t)
. (7.9)
This term does not vanish in the limit ν → 0 since the
solutions to the Burgers equation develop singularities
with a finite dissipation. It has been proposed in [99] to
use an analogy with the anomalies appearing in quantum
field theory in order to tackle this term in the inviscid
limit. The important assumption is then made that the
singular term in the operator product expansion relates
linearly to the characteristic function Zn. Since this ex-
pansion should preserve the statistical symmetries of the
Burgers equation, it leads to the replacement in all av-
erages of the singular limit limν→0 ν λ (∂
xv) e
λ v by the
asymptotic expression
b − 1
eλ v , (7.10)
where the coefficients a, b and c are parameters that can
be determined only indirectly. However their possible
values can be restricted by requiring that Zn is the char-
acteristic function of a probability distribution which is
non-negative, finite, normalizable, and that the dissipa-
tive term D(n)ν acts as a positive operator. Finding these
coefficients is similar to an eigenvalue problem in quan-
tum mechanics.
We now come to a crucial point in Polyakov’s approach.
Important restrictions on the form of the different
anomalous terms in (7.10) result from the fact that the
solutions to the Burgers equation obey a certain form of
Galilean invariance. A notion of “strong Galilean prin-
ciple” is introduced for invariance of the n-point distri-
bution of velocity under the transformation v 7→ v + v0
with v0 an arbitrary constant. As a consequence, the n-
point characteristic function Zn has to be proportional
to δ(λ1 + · · ·+λn). The operators appearing in the limit
ν → 0 have to be consistent with such an invariance.
In [99] it is argued that this symmetry is automatically
broken by the forcing that introduces a typical velocity
〈v2〉1/2 ∝ b1/3(0)L1/3. However Polyakov assumes this
“strong Galilean principle” to be asymptotically recov-
ered in the limit L → ∞ of infinite-size systems. In the
case of finite-size systems, when L is of the order of the
correlation length Lf of the forcing, the strong Galilean
symmetry is broken because of the conservation of the
spatial average of v which introduces a characteristic
velocity v0 = (1/L)
v(x, t) dx. However, the Galilean
symmetry should be recovered when averaging the cor-
relation functions with respect to the mean velocity v0.
This symmetry restoration was introduced in [20] where
it is referred to as the “weak Galilean principle”. The
n-point characteristic function associated to an aver-
age velocity v0 relates to that associated to a vanishing
mean velocity by
Zn(λj , xj ; t; v0) = e
Zn(λj , xj ; t; 0) .
After averaging with respect to v0, one obtains
Zn(λj , xj ; t) = 2π δ
 Zn(λj , xj ; t; 0) . (7.11)
One can easily check that (7.8), together with the dis-
sipative term given by (7.10), are compatible with this
expression for the n-point characteristic function Zn.
Moreover, any higher-order term in the expansion (7.10)
of the dissipative anomaly would violate Galilean invari-
ance.
To obtain the statistical properties of the solution, one
needs to further restrict the values of the three free pa-
rameters a, b, and c appearing in the expansion (7.10).
Following [99] this can be done by considering the case
n = 2 that corresponds to the equation for the PDF of
velocity differences. Performing the change of variables
λ1,2 = Λ ± µ and x1,2 = X ± y/2, and assuming that
λ ≪ µ and y ≪ Lf (so that the spatial part of the
forcing correlation is to leading order b(y) ≃ b0 − b1y2),
the stationary and space-homogeneous solutions to the
master equation (7.8)) satisfy
− (2b0Λ2 + b1µ2y2)Z2 =
= aZ2 +
. (7.12)
It is next assumed in [99] (see also [20]) that the velocity
difference v(x1, t) − v(x2, t) is statistically independent
of the mean velocity (v(x1, t) + v(x2, t))/2. This implies
that the two-point characteristic function factorizes as
Z2 = Z
2 (Λ)Z
2 (µ, y), where the two functions Z
2 and
Z−2 satisfy the closed equations
− 2b0Λ2Z+2 = cΛ
, (7.13)
∂2Z−2
− b1µ2y2Z−2 = aZ
. (7.14)
The solution to the first equation corresponds to a Gaus-
sian distribution which is normalizable only if c < 0. As
shown numerically in [20] this distribution is representa-
tive of the bulk of the one-point velocity PDF. Informa-
tion on the solutions to the second equation can be ob-
tained assuming the scaling property Z−2 (µ, y) = Φ(µy),
which amounts to considering only those contributions
to the distribution of velocity differences stemming from
velocity gradients ξ = ∂xv. This yields a prediction the
negative and positive tails of the PDF of velocity gradi-
ents:
p(ξ) ∝ |ξ|−α when ξ → −∞ , (7.15)
p(ξ) ∝ ξβ exp(−C ξ3) when ξ → +∞ , (7.16)
where C is a constant, which depends only on the
strength of the forcing. The two exponents α and β are
related to the coefficient b of the anomaly by
α = 2b + 1 and β = 2b − 1 . (7.17)
The value of b remains undetermined but is prescribed
to belong to a certain range. This approach was first
designed in [99] for infinite-size systems where strong
Galilean invariance holds. In that case consistency with
such an invariance leads to dropping the third term in
the operator product expansion (i.e. c = 0). Positivity
and normalizability of the two-point velocity PDF and
non-positivity of the anomalous dissipation operator im-
ply that the two other coefficients form a one-parameter
family with 3/4 ≤ b ≤ 1. In particular, this implies
that the left tail of the velocity gradient PDF with ex-
ponent α should be shallower than ξ−3. As we will see in
the next section, strong evidence has been obtained that
p(ξ) ∝ ξ−7/2 for ξ → −∞. This seems to contradict the
approach based on operator product expansion. How-
ever, as argued in [20], the breaking of strong Galilean in-
variance occurring in finite-size systems and resulting in
the presence of the c anomaly broadens the range of ad-
missible values for b. In particular it allows for the value
b = 5/4 which corresponds to the exponent α = 7/2.
7.3 Tails of the velocity gradient PDF
After the numerical work of Chekhlov and Yakhot [29],
the asymptotic behavior at large positive and negative
values of the PDF of velocity derivatives ξ = ∂xv for
the one-dimensional randomly forced Burgers equation
attracted much attention. A broad consensus emerged
around the prediction of Polyakov [99] that p(ξ) dis-
plays tails of the form (7.16) and (7.15), but the values
of the exponents α and β were at the center of a contro-
versy. Note that the presence of a super-exponential tail
∝ exp(−C ξ3) at large positive arguments has been con-
firmed by the use of instanton techniques [60] and that
the only remaining uncertainty concerns the exponent
of the algebraic prefactor. A standard approach to de-
termine the exponents α and β appearing in (7.15) and
(7.16) makes use of the stationary solutions to the in-
viscid limit of the Fokker–Planck equation for the PDF,
namely
−ξp+ν∂ξ
∂3xv | ∂xv=ξ
= b̃∂2ξp . (7.18)
Here the brackets 〈·|·〉 denote conditional averages and
the right-hand side expresses the diffusion of probability
due to the delta-correlation in time of the forcing. The
main difficulty in studying the solutions of (7.18) stems
from the treatment of the dissipative term Dν(ξ) =
∂3xv|∂xv=ξ
in the limit ν → 0. The value α = 3
is obtained if a piecewise linear approximation is made
for the solutions of the Burgers equation [21]. Gotoh and
Kraichnan [59] argued that the dissipative term is to
leading order negligible and presented analytical and nu-
merical arguments in favor of α = 3 and β = 1. However,
the inviscid limit of (7.18) contains anomalies due to the
singular behavior of Dν(ξ) in the limit ν → 0. As we
have seen in previous section, the approach based on the
use of an operator product expansion [99] leads to a rela-
tion involving unknown coefficients which must be deter-
mined, e.g., from numerical simulations [111,19,20], and
restricts the possible values to 5/2 ≤ α ≤ 3 [6]. Anoma-
lies cannot be studied without a complete description of
the singularities of the solutions, such as shocks, and a
thorough understanding of their statistical properties.
E, Khanin, Mazel and Sinai made a crucial observa-
tion in [37] that large negative gradients stem mainly
from preshocks, that is the cubic-root singularities in
the velocity preceding the formation of shocks (see sec-
tion 2.3). They then used a simple argument for de-
termining the fraction of space-time where the veloc-
ity gradient is less than some large negative value. This
leads to α = 7/2, provided preshocks do not cluster.
Later on, this approach has been refined by E and Van-
den Eijnden who proposed to determine the dissipative
anomaly of (7.18) using formal matched asymptotics [39]
or bounded variation calculus [42]. As we shall see be-
low, with the assumption that shocks are born with a
zero amplitude, that their strengths add up during colli-
sions, and that there ar no accumulations of preshocks,
the value α = 7/2 was confirmed [42]. Other attempts to
derive this value using also isolated preshocks have been
made [81,6]. Note that there are simpler instances, in-
cluding time-periodic forcing [9] (see section 6) and de-
caying Burgers turbulence with smooth random initial
conditions [8,42] (see section 4.1), which fall in the uni-
versality class α = 7/2, as can be shown by systematic
asymptotic expansions using a Lagrangian approach.
We give here the flavor of the approach used in [39]
in order to estimate the dissipative anomaly D0(ξ) =
limν→0D
ν(ξ). One first notices that for |ξ| ≫ b̃1/3, the
forcing term in the right-hand side of (7.18) becomes
negligible, so that stationary solutions to the Fokker–
Planck equation satisfy
p(ξ) ≈ |ξ|−3
dξ′ ξ′Dν(ξ′). (7.19)
A straightforward consequence of this asymptotic ex-
pression is that, if the integral in the right-hand side de-
creases as ξ → −∞ (i.e. if ξDν(ξ) is integrable), then
p(ξ) decreases faster than |ξ|−3, and thus α > 3.
To get some insight into the behavior ofDν as ν → 0, one
next observes that the solutions to the one-dimensional
Burgers equation contain smooth regions where viscos-
ity is negligible, which are separated by thin shock layers
where dissipation takes place. The basic idea consists in
splitting the velocity field v into the sum of an outer so-
lution away from shocks and of an inner solution near
them for which boundary layer theory applies. Matched
asymptotics are then used to construct a uniform ap-
proximation of v. To construct the inner solution near
a shock centered at x = x⋆, one performs the change of
variable x 7→ x̃ = (x−x⋆)/ν and looks for an expression
of ṽ(x̃, t) = v(x⋆+νx̃, t) in the form of a Taylor expan-
sion in powers of ν: ṽ = ṽ0+νṽ1+o(ν). At leading order,
the inner solution satisfies
[ṽ0 − v⋆] ∂x̃ṽ0 = ∂2x̃ṽ0, (7.20)
where v⋆ = (dx⋆)/(dt). This expression leads to the well-
known hyperbolic tangent velocity profile
ṽ0 = v⋆ −
. (7.21)
Here, s = v(x⋆+, t)−v(x⋆−, t) denotes here the velocity
jump across the shock and is given by matching condi-
tions to the outer solution. The term of order ν is then
a solution of
∂tṽ0 + [ṽ0 − v⋆] ∂x̃ṽ1 = ∂2x̃ṽ1 + f(x, t). (7.22)
In order to evaluate the dissipative anomaly, it is con-
venient to assume spatial ergodicity so that the viscous
term in (7.18) can be written as
Dν(ξ) = ν∂ξ lim
dx ∂3xv δ(∂xv−ξ). (7.23)
In the limit ν → 0 the only remaining contribution stems
from shocks and is thus given by the inner solution. Using
the expansion of the solution up to the first order in ν,
this leads to writing the dissipative term in the limit of
vanishing viscosity as (see Appendix of [41] for details)
D0(ξ) =
ds s [p+(s, ξ) + p−(s, ξ)] , (7.24)
where ρ is the density of shocks and p+ (respectively p−)
is the joint probability of the shock jump and of the value
of the velocity gradient at the right (respectively left) of
the shock. This expression guarantees the finiteness of
the dissipative anomaly, and in particular the fact that
the integral in the right-hand side of (7.19) is finite in
the limit ν → 0 and converges to 0. As a consequence,
this gives a proof that the exponent α of the left tail of
the velocity gradient PDF is larger than 3.
To proceed further, E and Vanden Eijnden proposed to
estimate the probability densities p+ and p− by deriving
master equations for the joint probability of the shock
strength s, its velocity v⋆ and the values ξ
± of the veloc-
ity gradient at its left and at its right. This is done in [42]
using a formulation of Burgers dynamics stemming from
bounded variation calculus. More precisely, it is shown
in [108] that the Burgers equation is equivalent to con-
sidering the solutions to the partial differential equation
∂tv + v̄∂xv = f , (7.25)
where v̄(x, t) = (v(x+, t) + v(x−, t))/2. Basically this
means that Burgers dynamics can be formulated in
terms of the transport of the velocity field by its average
v̄. This formulation straightforwardly yields a master
equation for v(x±, t) and ∂xv(x±, t) which is then used
to estimate p± and the dissipative anomaly (7.24). Al-
though the treatment of the master equation does not
involve any closure hypothesis, it is not fully rigorous:
in particular it requires the assumption that shocks are
created with zero amplitude and that shock amplitudes
add up during collision. However such an approaches
strongly suggests that α = 7/2 and β = 1.
Obtaining numerically clean scaling for the PDF of gra-
dients is not easy with standard schemes. Let us ob-
serve that any method involving a small viscosity, either
introduced explicitly (e.g. in a spectral calculation) or
stemming from discretization (e.g. in a finite difference
calculation), may lead to the presence of a power-law
range with exponent −1 at very large negative gradi-
ents [59]. This behavior makes the inviscid |ξ|−α range
appear shallower than it actually is, unless extremely
high spatial resolution is used. In contrast, methods that
directly capture the inviscid limit with the appropriate
shock conditions, such as the fast Legendre transform
method [94], lead to delicate interpolation problems.
They have been overcome in the case of time-periodic
forcing [9] but with white-noise-in-time forcing, it is dif-
ficult to prevent spurious accumulations of preshocks
leading to α = 3.
To avoid such pitfalls, a Lagrangian particle and shock
tracking method was developed in [6]. This method is
able to separate shocks and smooth parts of the solution
and is particularly effective for identifying preshocks.
The main idea is to consider the evolution of a set of
N massless point particles accelerated by a discrete-in-
time approximation of the forcing with a uniform time
step. When two of these particles intersect, they merge
and create a new type of particle, a shock, characterized
by its velocity (half sum of the right and left velocities
of merging particles) and its amplitude. The particle-
like shocks then evolve as ordinary particles, capture
further intersecting particles and may merge with other
shocks. In order not to run out of particles too quickly,
the initial small region where particles have the least
chance of being subsequently captured is determined by
localization of the global minimizer of the Lagrangian
action (see section 5.1). The calculation is then restarted
from t = 0 for the same realization of forcing but with
a vastly increased number of particles in that region.
This particle and shock tracking method gives complete
control over shocks and preshocks.
slope = −7/2
−3.75
−3.25
−7/2±1%
Fig. 40. PDF of the velocity gradient at negative values in
log-log coordinates obtained by averaging over 20 realiza-
tions and a time interval of 5 units of time (after relaxation
of transients). The simulation involves up to N = 105 parti-
cles and the forcing is applied at discrete times separated by
δt = 10−4. Upper inset: local scaling exponent (from [6]).
Figure 40 shows the PDF of the velocity gradients in
log-log coordinates at negative values, for a Gaussian
forcing restricted to the first three Fourier modes with
equal variances such that the large-scale turnover time is
order unity. Quantitative information about the value of
the exponent is obtained by measuring the “local scaling
exponent”, i.e. the logarithmic derivative of the PDF
calculated in this case using least-square fits on half-
decades. It is seen that over about five decades, the local
exponent is within less than 1% of the value α = 7/2
predicted by E et al. [37].
7.4 Self-similar forcing and multiscaling
As we have seen in section 7.1, the solutions to the Burg-
ers equation in a finite domain and with a large-scale
forcing have structure functions (moments of the ve-
locity increment) displaying a bifractal scaling behav-
ior. Such a property can be easily interpreted by the
presence of a finite number of shocks with a size order
unity in the finite system. Somehow this double scal-
ing and its relationship with singularities gives some in-
sight on the multiscaling properties that are expected
in the case of turbulent incompressible hydrodynamics
flows. There is a general consensus that the turbulent
solutions to the Navier–Stokes equations display a full
multifractal spectrum of singularities which are respon-
sible for a nonlinear p-dependence of the scaling expo-
nents ζp associated to the scaling behavior of the p-th
order structure function [53]. The construction of simple
tractable models which are able to reproduce such a be-
havior has motivated much work during the last decades.
Significant progress, both analytical and numerical, has
been made in confirming multiscaling in passive-scalar
and passive-vector problems (see, e.g., [46] for a review).
However, the linearity of the passive-scalar and passive-
vector equations is a crucial ingredient of these studies,
so it is not clear how they can be generalized to fluid
turbulence and the Navier–Stokes equation.
After the work of Chekhlov and Yakhot [30], it appeared
that the Burgers equation with self-similar forcing could
be the simplest nonlinear partial differential equation
which has the potential to display multiscaling of veloc-
ity structure functions. We report in this section various
works that tried to confirm or to weaken this statement.
Let us consider the solutions to the one-dimensional
Burgers equation with a forcing term f(x, t) which is ran-
dom, space-periodic, Gaussian and whose spatial Fourier
transform has correlation
〈f̂(k, t)f̂(k′, t′)〉 = 2D0 |k|β δ(t− t′) δ(k + k′). (7.26)
The exponent β determines the scaling properties of the
forcing. When β > 0 the force acts at small scales; for
instance β = 2 corresponds to thermal noise for the ve-
locity potential, and thus to the KPZ model for interface
growth [74]. It is well known in this case (see, e.g., [5])
that the solution displays simple scaling (usually known
as KPZ scaling), such that ζq = q for all q. More gener-
ally, the case β > 0 can be exactly solved using a one-
loop renormalization group approach [88].
As stressed in [64], renormalization group techniques fail
when β < 0 and the forcing acts mostly at large scales
and non-linear terms play a crucial role. When β < −3,
the forcing is differentiable in the space variable, the so-
lution is piecewise smooth and contains a finite number
of shocks with sizes order unity. The scaling exponents
are then ζp = min (1, p). In the case of non-differentiable
forcing (−3 < β < 0), the presence of order-unity shocks
and dimensional arguments suggest that the scaling ex-
ponents are ζp = min (1, −pβ/3). However, very little is
known regarding the distribution of shocks with inter-
mediate sizes. In particular, there is no clear evidence
whether or not they form a self-similar structure at small
scales. We summarize here some studies which were done
on Burgers turbulence with self-similar forcing to show
how difficult it might be to measure scaling laws of struc-
ture functions and in particular how logarithmic correc-
tions can masquerade anomalous scaling.
For this we focus on the case β = −1 which has attracted
much attention; indeed, dimensional analysis suggests
that ζp = p/3 when p ≤ 3, leading to a K41-type −5/3
energy spectrum. Early studies [29,30] seemed to confirm
this prediction using pseudo-spectral viscous numerical
Fig. 41. Representative snapshots of the velocity v (jagged
line) in the statistically stationary régime, and of the inte-
gral of the force f over a time step (rescaled for plotting
purposes).
simulations at rather low resolutions (around ten thou-
sands gridpoints). It was moreover argued in [64,65] that
a self-similar forcing with −1 < β < 0, could lead to gen-
uine multifractality. The lack of accuracy in the determi-
nation of the scaling exponents left open the question of
a weak anomalous deviation from the dimensional pre-
diction. This question was recently revisited in [91] with
high-resolution inviscid numerical simulations using the
fast Legendre transform algorithm (see section 2.4.2).
A typical snapshot of the forcing and of the solution in
the stationary régime are represented in figure 41. It is
clear that because of shocks the velocity develops small-
scale fluctuations much stronger than those present in
the force. However one notices that shock dynamics and
spatial finiteness of the system lead, as predicted, to the
presence of few shocks with order-unity sizes.
0 1 2 3 4 5
ξ p 0 1 2 3 4 5
Fig. 42. Scaling exponents ζp versus order p for
N = 216(⋄), 218(∗), and 220(◦) grid points. Error bars (see
text) are shown for the case N = 220. The deviation of ζp
from the exponents for bifractal scaling (full lines), shown
as an inset, naively suggests multiscaling (from [91])
Structure functions were measured with high accuracy.
They typically exhibit a power-law behavior over nearly
three decades in length scale; this is more than two
decades better than in [30]. In principle one expects to
be able to measure the scaling exponents with enough
accuracy to decide between bifractality and multiscal-
ing. Surprisingly the naive analysis summarized in fig-
ure 42 does suggest multiscaling: the exponents ζp de-
viate significantly from the bifractal-scaling prediction
(full lines). Since the goal here is to have a precise han-
dle on the scaling properties of velocity increments, it is
important to carefully define how the scaling exponents
are measured. They are estimated from the average log-
arithmic derivative of Sabsp (r) = 〈|v(x+r)−v(x)|p〉 over
almost two decades in the separation r. The error bars
shown are given by the maximum and minimum devia-
tions from this mean value in the fitting range. Note also
that the observed multiscaling is supported by the fact
that there is no substantial change in the value of the
exponents when changing the number N of grid points
in the simulation from 216 to 220: any dependence of
ζp upon N is much less than the error bars determined
through the procedure described above.
−5 −4 −3 −2 −1
−6 −4 −2
Fig. 43. Log-log plots of Sabs3 (r) (dashed line), S3(r)
(crosses), and 〈(δ+v)3〉 (squares) versus r. The continuous
line is a least-square fit to the range of points limited by two
vertical dashed lines in the plot. Inset: An explicit check of
the von Kármán–Howarth relation (7.27) from the simula-
tions with N = 220 reported in [91]. The dashed curve is the
integral of the spatial part of the forcing correlation and the
circles represent the numerical computation of the left-hand
side.
As found in [91], the observed deviations of the scaling
exponents from bifractality are actually due to the con-
tamination by subleading terms in Sabsp (r). To quantify
this effect, let us focus on the third-order structure func-
tion (p = 3) for which one measures ζ3 ≈ 0.85±0.02 over
nearly four decades (see figure 43). To estimate sublead-
ing terms we first notice that the third-order structure
function S3(r) ≡ 〈(v(x + r) − v(x))3〉, which is defined,
this time, without the absolute value, obeys an analog of
the von Kármán–Howarth relation in fluid turbulence,
namely
S3(r) =
b(r′)dr′, (7.27)
where b(·) denotes the spatial part of the force corre-
lation function, i.e. 〈f(x + r, t′)f(x, t)〉 = b(r)δ(t − t′).
This relation, together with the correlation (7.26) and
β = −1, implies the behavior S3(r) ∼ r ln r at small
separations r. As seen in figure 43, the graph of S3(r)
in log-log coordinates indeed displays a significant cur-
vature which is a signature of logarithmic corrections.
The next step consists in decomposing the velocity in-
crements δrv = v(x + r, t) − v(x, t) into their positive
δ+r v and negative δ
r v parts. It is clear that
Sabs3 (r) = −
(δ−r v)
(δ+r v)
S3(r) =
(δ−r v)
(δ+r v)
, (7.28)
so that
(δ+r v)
= (Sabs3 (r) + S3(r))/2. As seen in fig-
ure 43 the log-log plot of
(δ+r v)
as a function of r
is nearly a straight line with slope ≈ 1.07 very close to
unity. This observation is confirmed in [91] by indepen-
dently measuring the PDFs of positive and negative ve-
locity increments. Assuming that
(δ+r v)
∼ B r, one
obtains the following prediction for the small-r behav-
iors of the third-order structure functions
Sabs3 (r) ∼ −Ar ln r +B r,
S3(r) ∼ Ar ln r +B r. (7.29)
This suggests that the only difference in the small-
separation behaviors of Sabs3 (r) and S3(r) is the sign in
the balance between the leading term ∝ r ln r and the
subleading term ∝ r. In a log-log plot this difference
amounts to shifting the graph away from where it is
most curved and thus makes it straighter, albeit with a
(local) slope which is not unity. This explains why signif-
icant deviations from 1 are observed for ζ3. Note that a
similar approach can be used for higher-order structure
functions. It leads for instance to S4(r) ≈ Cr −Dr4/3,
where C and D are two positive constants. The nega-
tive sign before the sub-leading term (r4/3) is crucial.
It implies that, for any finite r, a naive power-law fit
to S4 can yield a scaling exponent less than unity. The
presence of sub-leading, power-law terms with oppo-
site signs also explains the small apparent “anomalous”
scaling behavior observed for other values of p in the
simulations. Note that similar artifacts involving two
competing power-laws have been described in [16,7].
The work reported in this section indicates that a naive
interpretation of numerical measurements might result
in predicting artificial anomalous scaling laws. In the
case of Burgers turbulence for which high-resolution nu-
merics are available and statistical convergence of the
averages can be guaranteed, we have seen that it is not
too difficult to identify the numerical artifacts which
are responsible for such a masquerading. However this
is not always the case. For instance, it seems reason-
able enough to claim that attacking the problem of mul-
tiscaling in spatially extended nonlinear systems, such
as Navier–Stokes turbulence, requires considerable the-
oretical insight that must supplement sophisticated and
heavy numerical simulations and experiments. Note fi-
nally that, up to now, the question of the presence or not
of anomalous scaling laws in the Burgers equation with
a self-similar forcing with exponent −1 < β < 0 remains
largely open.
8 Concluding remarks and open questions
This review summarizes recent work connected with the
Burgers equation. Originally this model was introduced
as a simplification of the Navier–Stokes equation with
the hope of shedding some light on issues such as turbu-
lence. This hope did not materialize. Nevertheless many
of the interesting questions that have been addressed
for Burgers turbulence are eventually transpositions of
similar questions for Navier–Stokes turbulence. One par-
ticularly important instance is the issue of universality
with respect to the form of the forcing and of the initial
condition. For Burgers turbulence most of the universal
features, such as scaling exponents or functional forms
of PDF tails are dominated by the presence of shocks
and other singularities in the solution. This applies both
to the case of decaying turbulence driven by random ini-
tial conditions and randomly forced turbulence. In the
latter case one is mostly interested in analysis of station-
ary properties of solutions, for example stationary dis-
tribution for velocity increments or gradients. Another
set of questions is motivated by more mathematical con-
siderations. It mainly concerns the construction of a sta-
tionary invariant measure when Burgers dynamics in a
finite-size domain is supplemented by an external ran-
dom source of energy. Again it has been shown that the
presence of shocks, and in particular of global shocks,
plays a crucial role in the construction of the statisti-
cally stationary solution. Both physical and mathemat-
ical questions lead to a similar answer: one first needs to
describe and control shocks. The main message to retain
for hydrodynamical turbulence is hence a strong confir-
mation of the common wisdom that it cannot be fully
understood without a detailed description of singulari-
ties. Moreover, the behavior depends not only on the lo-
cal structure of singularities, but also on their distribu-
tion at larger scales. Here a word of caution: for incom-
pressible fully developped Navier–Stokes turbulence, we
have no evidence that the universal scaling properties
observed in experiments and simulations stem from real
singularities. Indeed the issue of a finite-time blow-up of
the three-dimensional Euler equation is still open (see,
e.g. [56]). Another important observation that can be
drawn from the study of Burgers turbulence is that both
the tools used and the answers obtained strongly de-
pend on the kind of setting one considers: decay versus
forced turbulence, finite-size versus infinite-size systems,
smooth versus self-similar forcing, etc.
Besides turbulence, the random Burgers equation has
various applications in cosmology, in non-equilibrium
statistical physics and in disordered media. Among
them, the connection to the problem of directed poly-
mers has attracted much attention. As already noted in
the Introduction, there is a mathematical equivalence
between the zero-viscosity limit of the forced Burgers
equation and the zero-temperature limit for directed
polymers. We have seen in section 5.4 that the so-called
KPZ scaling, which usually is derived for a finite tem-
perature, can be established can be established also in
the zero-temperatur limit, using the action minimizer
representation. Such an observation leads to two related
questions: to what extent can the limit of zero temper-
ature give an insight into finite-temperature polymer
dynamics and how can the global minimizer formalism
be extended to tackle the finite-temperature setting?
It looks plausible that in polymer dynamics, or more
generally in the study of random walks in a random
potential, the trajectories carrying most of the Gibbs
probability weight are defining corridors in space time.
These objects can concentrate near the trajectories of
global minimizers but, at the moment, there is no for-
malism to describe them, nor attempts to quantify their
contribution to the Gibbs statistics.
Another important open question concerns the multi-
dimensional extensions of the Burgers equation. As we
have seen, when the forcing is potential, the potential
character of the velocity field is conserved by the dynam-
ics. This leads to the construction of stationary solutions
which carry many similarities with the one-dimensional
case. Up to now there is only limited understanding of
what happens when the potentiality assumption of the
flow is dropped. This problem has of course concrete
applications in gas dynamics and for disperse inelastic
granular media (see, e.g., [12]). An interesting question
concerns the construction of the limit of vanishing vis-
cosity, given that the Hopf–Cole transformation is in-
applicable in the non-potential case. Understanding ex-
tensions of the viscous limiting procedure to the non-
potential case might give new insight into the problem of
the large Reynolds number limit in incompressible tur-
bulence. Another question related to non-potential flow
concerns the interactions between vorticity and shocks.
For instance, in two dimensions the vorticity is trans-
ported by the flow. This results in its growth in the highly
compressible regions of the flow. The various singulari-
ties of the velocity field should hence be strongly affected
by the flow rotation and, in particular, the shocks are
expected to have a spiraling structure.
We finish with few remarks on open mathematical prob-
lems. As we have seen in the one-dimensional case, one
can rigorously prove hyperbolicity of the global mini-
mizer. In the multi-dimensional case it is also possible
to establish the existence and, in many cases, unique-
ness of the global minimizer. However, the very impor-
tant question of its hyperbolicity is still an open prob-
lem. If proven, hyperbolicity would allow for rigorous
analysis of the regularity properties of the stationary so-
lutions and of the topological shocks. There are many
interesting problems – even basic issues of existence and
uniqueness – in the non-compact case where at present a
mathematical theory is basically absent. Finally, a very
challenging open problem concerns the extension of the
results on the evolution of matter inside shocks to the
case of general Hamilton-Jacobi equations.
Acknowledgements
Over the years of our work on Burgers turbulence, we
profited a lot from numerous discussions with Uriel
Frisch whose influence on our work is warmly acknowl-
edged. We also want to express our sincere gratitude
to all of our collaborators: W. E, U. Frisch, D. Gomes,
V.H. Hoang, R. Iturriaga, D. Khmelev, A. Mazel, D. Mi-
tra, P. Padilla, R. Pandit, Ya. Sinai, A. Sobolevskĭı, and
B. Villone.
While writing this article, we benefited from discus-
sions with M. Blank, I. Bogaevsky, K. Domelevo, V.
Epstein, and A. Sobolevskĭı. Finally, our thanks go to
Itamar Procaccia whose encouragements and patience
are greatly appreciated.
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	From interface dynamics to cosmology
	The Burgers equation in statistical mechanics
	The adhesion model in cosmology
	A benchmark for hydrodynamical turbulence
	Basic tools
	Inviscid limit and variational principle
	Variational principle for the viscous case
	Singularities of Burgers turbulence
	Remarks on numerical methods
	Decaying Burgers turbulence
	Geometrical constructions of the solution
	Kida's law for energy decay
	Brownian initial velocities
	Transport of mass in the Burgers/adhesion model
	Mass density and singularities
	Evolution of matter inside shocks
	Connections with convex optimization problems
	Forced Burgers turbulence
	Stationary régime and global minimizer
	Topological shocks
	Hyperbolicity of the global minimizer
	The case of extended systems
	Time-periodic forcing
	Kicked Burgers turbulence
	Connections with Aubry--Mather theory
	Velocity statistics in randomly forced Burgers turbulence
	Shocks and bifractality -- a replica variational approach
	Dissipative anomaly and operator product expansion
	Tails of the velocity gradient PDF
	Self-similar forcing and multiscaling
	Concluding remarks and open questions
	Acknowledgements
	References
