Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains
Bursting Dynamics of the 3D Euler Equations
in Cylindrical Domains
François Golse ∗ †
Ecole Polytechnique, CMLS
91128 Palaiseau Cedex, France
Alex Mahalov ‡and Basil Nicolaenko §
Department of Mathematics and Statistics
Arizona State University
Tempe, AZ 85287-1804, USA
Abstract
A class of three-dimensional initial data characterized by uniformly
large vorticity is considered for the 3D incompressible Euler equations in
bounded cylindrical domains. The fast singular oscillating limits of the
3D Euler equations are investigated for parametrically resonant cylin-
ders. Resonances of fast oscillating swirling Beltrami waves deplete the
Euler nonlinearity. These waves are exact solutions of the 3D Euler
equations. We construct the 3D resonant Euler systems; the latter are
countable uncoupled and coupled SO(3;C) and SO(3;R) rigid body
systems. They conserve both energy and helicity. The 3D resonant Eu-
ler systems are vested with bursting dynamics, where the ratio of the
enstrophy at time t = t∗ to the enstrophy at t = 0 of some remarkable
orbits becomes very large for very small times t∗; similarly for higher
norms Hs, s ≥ 2. These orbits are topologically close to homoclinic
cycles. For the time intervals where Hs norms, s ≥ 7/2 of the limit
resonant orbits do not blow up, we prove that the full 3D Euler equa-
tions possess smooth solutions close to the resonant orbits uniformly in
strong norms.
Key-Words: Incompressible Euler Equations, Rotating Fluids, Rigid Body
Dynamics, Enstrophy Bursts
MSC: 35Q35, 76B03, 76U05
∗golse@math.polytechnique.fr
†and Laboratoire J.-L. Lions, Université Paris Diderot-Paris 7
‡mahalov@asu.edu
§byn@stokes.la.asu.edu
http://arxiv.org/abs/0704.0337v1
1 Introduction
The issues of blowup of smooth solutions and finite time singularities of
the vorticity field for 3D incompressible Euler equations are still a major
open problem. The Cauchy problem in 3D bounded axisymmetric cylindrical
domains is attracting considerable attention: with bounded, smooth, non-
axisymmetric 3D initial data, under the constraints of conservation of bounded
energy, can the vorticity field blow up in finite time? Outstanding numeri-
cal claims for this have recently been disproven [Ke], [Hou1], [Hou2]. The
classical analytical criterion of Beale-Kato-Majda [B-K-M] for non-blow up
in finite time requires the time integrability of the L∞ norm of the vorticity.
DiPerna and Lions [Li] have given examples of global weak solutions of the
3D Euler equations which are smooth (hence unique) if the initial conditions
are smooth (specifically in W1,p(D), p > 1). However, these flows are really
2-Dimensional in x1, x2, 3-components flows, independent from the third co-
ordinate x3. Their examples [DiPe-Li] show that solutions (even smooth ones)
of the 3D Euler equations cannot be estimated in W1,p for 1 < p <∞ on any
time interval (0, T ) if the initial data are only assumed to be bounded inW1,p.
Classical local existence theorems in 3D bounded or periodic domains by Kato
[Ka], Bourguignon-Brézis [Bou-Br] and Yudovich [Yu1], [Yu2] require some
minimal smoothness for the initial conditions (IC), e.g., in Hs(D), s > 5
The classical formulation for the Euler equations is
∂tV+ (V · ∇)V = −∇p, ∇ ·V = 0, (1.1)
V ·N = 0 on ∂D, (1.2)
where ∂D is the boundary of a bounded, connected domain D, N the normal
to ∂D, V(t, y) = (V1, V2, V3) the velocity field, y = (y1, y2, y3), and p is the
pressure.
The equivalent Lamé form [Ar-Khe]
∂tV + curlV ×V +∇
= 0, (1.3)
∇ ·V = 0, (1.4)
∂tω + curl(ω ×V) = 0, (1.5a)
ω = curlV, (1.5b)
implies conservation of Energy:
E(t) =
|V(t, y)|2 dy. (1.6)
The helicity Hel(t) [Ar-Khe], [Mof], is conserved:
Hel(t) =
V · ω dy, (1.7)
for D = R3 and when D is a periodic lattice. Helicity is also conserved for
cylindrical domains, provided that ω·N = 0 on the cylinder’s lateral boundary
at t = 0 (see [M-N-B-G]).
From the theoretical point of view, the principal difficulty in the analysis
of 3D Euler equations is due to the presence of the vortex stretching term
(ω · ∇)V in the vorticity equation (1.5a). The equations (1.3) and (1.5a) are
equivalent to:
∂tω + [ω,V] = 0, (1.8)
where [a, b] = curl (a × b) is the commutator in the infinite dimensional Lie
algebra of divergence-free vector fields [Ar-Khe]. This point of view has led to
celebrated developments in Topological Methods in Hydrodynamics [Ar-Khe],
[Mof]. The striking analogy between the Euler equations for hydrodynamics
and the Euler equations for a rigid body (the latter associated to the Lie
Algebra of the Lie group SO(3,R)) had already been pointed out by Moreau
[Mor1]; Moreau was the first to demonstrate conservation of Helicity (1961)
[Mor2]. This has led to extensive speculations to what extent/in what cases
are the solutions of the 3D Euler equations “close” to those of coupled 3D rigid
body equations in some asymptotic sense. Recall that the Euler equations for
a rigid body in R3 is:
mt + ω ×m = 0, m = Aω, (1.9a)
mt + [ω,m] = 0, (1.9b)
where m is the vector of angular momentum relative to the body, ω the
angular velocity in the body and A the inertia operator [Ar1], [Ar-Khe].
The Russian school of Gledzer, Dolzhansky, Obukhov [G-D-O] and Vishik
[Vish] has extensively investigated dynamical systems of hydrodynamic type
and their applications. They have considered hydrodynamical models built
upon generalized rigid body systems in SO(n,R), following Manakhov [Man].
Inspired by turbulence physics, they have investigated “shell” dynamical sys-
tems modeling turbulence cascades; albeit such systems are flawed as they
only preserve energy, not helicity. To address this, they have constructed and
studied in depth n-dimensional dynamical systems with quadratic homoge-
neous nonlinearities and two quadratic first integrals F1, F2. Such systems
can be written using sums of Poisson brackets:
i2,...,in
ǫi1i2...inpi4...in
− ∂F1
, (1.10)
where constants pi4...in are antisymmetric in i4, ..., in.
A simple version of such a quadratic hydrodynamic system was introduced
by Gledzer [Gl1] in 1973. A deep open issue of the work by the Gledzer-
Obukhov school is whether there exist indeed classes of I.C. for the 3D Cauchy
Euler problem (1.1) for which solutions are actually asymptotically close in
strong norm, on arbitrary large time intervals to solutions of such hydro-
dynamic systems, with conservation of both energy and helicity. Another
unresolved issue is the blowup or global regularity for the “enstrophy” of such
systems when their dimension n→ ∞.
This article reviews some current new results of a research program in
the spirit of the Gledzer-Obukhov school; this program builds-up on the re-
sults of [M-N-B-G] for 3D Euler in bounded cylindrical domains. Following
the original approach of [B-M-N1]-[B-M-N4] in periodic domains, [M-N-B-G]
prove the non blowup of the 3D incompressible Euler equations for a class
of three-dimensional initial data characterized by uniformly large vorticity in
bounded cylindrical domains. There are no conditional assumptions on the
properties of solutions at later times, nor are the global solutions close to
some 2D manifold. The initial vortex stretching is large. The approach of
proving regularity is based on investigation of fast singular oscillating limits
and nonlinear averaging methods in the context of almost periodic functions
[Bo-Mi], [Bes], [Cor]. Harmonic analysis tools based on curl eigenfunctions
and eigenvalues are crucial. One establishes the global regularity of the 3D
limit resonant Euler equations without any restriction on the size of 3D initial
data. The resonant Euler equations are characterized by a depleted nonlin-
earity. After establishing strong convergence to the limit resonant equations,
one bootstraps this into the regularity on arbitrary large time intervals of the
solutions of 3D Euler Equations with weakly aligned uniformly large vorticity
at t = 0. [M-N-B-G] theorems hold for generic cylindrical domains, for a set
of height/radius ratios of full Lebesgue measure. For such cylinders, the 3D
limit resonant Euler equations are restricted to two-wave resonances of the
vorticity waves and are vested with an infinite countable number of new con-
servation laws. The latter are adiabatic invariants for the original 3D Euler
equations.
Three-wave resonances exist for a nonempty countable set of h/R (h
height, R radius of the cylinder) and moreover accumulate in the limit of
vanishingly small vertical (axial) scales. This is akin to Arnold tongues [Ar2]
for the Mathieu-Hill equations and raises nontrivial issues of possible sin-
gularities/lack thereof for dynamics ruled by infinitely many resonant triads
at vanishingly small axial scales. In such a context, the 3D resonant Euler
equations do conserve the energy and helicity of the field.
In this review, we consider cylindrical domains with parametric resonances
in h/R and investigate in depth the structure and dynamics of 3D resonant
Euler systems. These parametric resonances in h/R are proven to be non-
empty. Solutions to Euler equations with uniformly large initial vorticity are
expanded along a full complete basis of elementary swirling waves (T2 in time).
Each such quasiperiodic, dispersive vorticity wave is a quasiperiodic Beltrami
flow; these are exact solutions of 3D Euler equations with vorticity parallel
to velocity. There are no Galerkin-like truncations in the decomposition of
the full 3D Euler field. The Euler equations, restricted to resonant triplets of
these dispersive Beltrami waves, determine the “resonant Euler systems”. The
basic “building block” of these (a priori ∞-dimensional) systems are proven
to be SO(3;C) and SO(3;R) rigid body systems:
U̇k + (λm − λn)UmUn = 0
U̇m + (λn − λk)UnUk = 0
U̇n + (λk − λm)UkUm = 0
(1.11)
These λ’s are eigenvalues of the curl operator in the cylinder, curlΦ±n =
±λnΦ±n ; the curl eigenfunctions are steady elementary Beltrami flows, and
the dispersive Beltrami waves oscillate with the frequencies ± h
, n3 ver-
tical wave number (vertical shear), 0 < ǫ < 1. Physicists [Ch-Ch-Ey-H] have
computationally demonstrated the physical impact of the polarization of Bel-
trami modes Φ± on intermittency in the joint cascade of energy and helicity
in turbulence.
Another “building block” for resonant Euler systems is a pair of SO(3;C)
or SO(3;R) rigid bodies coupled via a common principal axis of inertia/mo-
ment of inertia:
ȧk = (λm − λn)Γaman (1.12a)
ȧm = (λn − λk)Γanak (1.12b)
ȧn = (λk − λm)Γakam + (λk̃ − λm̃)Γ̃ak̃am̃ (1.12c)
ȧm̃ = (λn − λk̃)Γ̃anak̃ (1.12d)
ȧk̃ = (λm̃ − λn)Γ̃am̃an, (1.12e)
where Γ and Γ̃ are parameters in R defined in Theorem 4.10. Both reso-
nant systems (1.11) and (1.12) conserve energy and helicity. We prove that
the dynamics of these resonant systems admit equivariant families of homo-
clinic cycles connecting hyperbolic critical points. We demonstrate bursting
dynamics: the ratio
||u(t)||2Hs/||u(0)||2Hs , s ≥ 1
can burst arbitrarily large on arbitrarily small times, for properly chosen para-
metric domain resonances h/R. Here
||u(t)||2Hs =
2s|un(t)|2 . (1.13)
The case s = 1 is the enstrophy. The “bursting” orbits are topologically close
to the homoclinic cycles.
Are such dynamics for the resonant systems relevant to the full 3D Euler
equations (1.1)-(1.8)? The answer lies in the following crucial “shadowing”
Theorem 2.10. Given the same initial conditions, given the maximal time
interval 0 ≤ t < Tm where the resonant orbits of the resonant Euler equations
do not blow up, then the strong norm Hs of the difference between the exact
Euler orbit and the resonant orbit is uniformly small on 0 ≤ t < Tm, provided
that the vorticity of the I.C. is large enough. Paradoxically, the larger the
vortex streching of the I.C., the better the uniform approximation. This deep
result is based on cancellation of fast oscillations in strong norms, in the
context of almost periodic functions of time with values in Banach spaces
(Section 4 of [M-N-B-G]). It includes uniform approximation in the spaces
Hs, s > 5/2. For instance, given a quasiperiodic orbit on some time torus Tl
for the resonant Euler systems, the exact solutions to the Euler equations will
remain ǫ-close to the resonant quasiperiodic orbit on a time interval 0 ≤ t ≤
maxTi, 1 ≤ i ≤ l, Ti elementary periods, for large enough initial vorticity. If
orbits of the resonant Euler systems admit bursting dynamics in the strong
norms Hs, s ≥ 7/2, so do some exact solutions of the full 3D Euler equations,
for properly chosen parametrically resonant cylinders.
2 Vorticity waves and resonances of elemen-
tary swirling flows
We study initial value problem for the three-dimensional Euler equations with
initial data characterized by uniformly large vorticity:
∂tV+ (V · ∇)V = −∇p, ∇ ·V = 0, (2.1)
V(t, y)|t=0 = V(0) = Ṽ0(y) +
e3 × y (2.2)
where y = (y1, y2, y3), V(t, y) = (V1, V2, V3) is the velocity field and p is the
pressure. In Eqs. (1.1) e3 denotes the vertical unit vector and Ω is a constant
parameter. The field Ṽ0(y) depends on three variables y1, y2 and y3. Since
curl(Ω
e3 × y) = Ωe3, the vorticity vector at initial time t = 0 is
curlV(0, y) = curlṼ0(y) + Ωe3, (2.3)
and the initial vorticity has a large component weakly aligned along e3, when
Ω >> 1. These are fully three-dimensional large initial data with large initial
3D vortex stretching. We denote by Hsσ the usual Sobolev space of solenoidal
vector fields.
The base flow
Vs(y) =
e3 × y, curlVs(y) = Ωe3 (2.4)
is called a steady swirling flow and is a steady state solution (1.1)-(1.4), as
curl(Ωe3×Vs(y)) = 0. In (2.2) and (2.3), we consider I.C. which are an arbi-
trary (not small) perturbation of the base swirling flow Vs(y) and introduce
V(t, y) =
e3 × y + Ṽ(t, y), (2.5)
curlV(t, y) = Ωe3 + curlṼ(t, y), (2.6)
∂tṼ + curlṼ× Ṽ +Ωe3 × Ṽ + curlṼ×Vs(y) +∇p′ = 0, ∇ · Ṽ = 0, (2.7)
Ṽ(t, y)|t=0 = Ṽ0(y). (2.8)
Eqs. (2.1) and (2.7) are studied in cylindrical domains
C = {(y1, y2, y3) ∈ R3 : 0 < y3 < 2π/α, y21 + y22 < R2} (2.9)
where α and R are positive real numbers. If h is the height of the cylinder,
α = 2π/h. Let
Γ = {(y1, y2, y3) ∈ R3 : 0 < y3 < 2π/α, y21 + y22 = R2}. (2.10)
Without loss of generality, we can assume that R = 1. Eqs. (2.1) are consid-
ered with periodic boundary conditions in y3
V(y1, y2, y3) = V(y1, y2, y3 + 2π/α) (2.11)
and vanishing normal component of velocity on Γ
V ·N = Ṽ ·N = 0 on Γ; (2.12)
where N is the normal vector to Γ. From the invariance of 3D Euler equations
under the symmetry y3 → −y3, V1 → V1, V2 → V2, V3 → −V3, all results in
this article extend to cylindrical domains bounded by two horizontal plates.
Then the boundary conditions in the vertical direction are zero flux on the
vertical boundaries (zero vertical velocity on the plates). One only needs to
restrict vector fields to be even in y3 for V1, V2 and odd in y3 for V3, and
double the cylindrical domain to −h ≤ y3 ≤ +h.
We choose Ṽ0(y) in H
s(C), s > 5/2. In [M-N-B-G], for the case of “non-
resonant cylinders”, that is, non-resonant α = 2π/h, we have established
regularity for arbitrarily large finite times for the 3D Euler solutions for Ω
large, but finite. Our solutions are not close in any sense to those of the 2D
or “quasi 2D” Euler and they are characterized by fast oscillations in the e3
direction, together with a large vortex stretching term
ω(t, y) · ∇V(t, y) = ω1
, t ≥ 0
with leading component
V(t, y)
≫ 1. There are no assumptions on
oscillations in y1, y2 for our solutions (nor for the initial condition Ṽ0(y)).
Our approach is entirely based on sturying fast singular oscillating limits
of Eqs. (1.1)-(1.5a), nonlinear averaging and cancelation of oscillations in the
nonlinear interactions for the vorticity field for large Ω. This has been devel-
oped in [B-M-N2], [B-M-N3], and [B-M-N4] for the cases of periodic lattice
domains and the infinite space R3.
It is well known that fully three-dimensional initial conditions with uni-
formly large vorticity excite fast Poincaré vorticity waves [B-M-N2], [B-M-N3],
[B-M-N4], [Poi]. Since individual Poincaré wave modes are related to the
eigenfunctions of the curl operator, they are exact time-dependent solutions
of the full nonlinear 3D Euler equations. Of course, their linear superposition
does not preserve this property. Expanding solutions of (2.1)-(2.8) along such
vorticity waves demonstrates potential nonlinear resonances of such waves.
First recall spectral properties of the curl operator in bounded, connected
domains:
Proposition 2.1 ([M-N-B-G]) The curl operator admits a self-adjoint ex-
tension under the zero flux boundary conditions, with a discrete real spectrum
λn = ±|λn|, |λn| > 0 for every n and |λn| → +∞ as |n| → ∞. The corre-
sponding eigenfunctions Φ±n
curlΦ±n = ±|λn|Φ
n (2.13)
are complete in the space
U ∈ L2(D) : ∇ ·U = 0 and U ·N|∂D = 0 and
U dz = 0
(2.14)
Remark 2.2 In cylindrical domains, with cylindrical coordinates (r, θ, z), the
eigenfunctions admit the representation:
Φn1,n2,n3 = (Φr,n1,n2,n3(r),Φθ,n1,n2,n3(r),Φz,n1,n2,n3(r)) e
in2θeiαn3z, (2.15)
with n2 = 0,±1,±2, ..., n3 = ±1,±2, ... and n1 = 0, 1, 2, .... Here n1 indexes
the eigenvalues of the equivalent Sturm-Liouville problem in the radial coor-
dinates, and n = (n1, n2, n3). See [M-N-B-G] for technical details. From now
on, we use the generic variable z for any vertical (axial) coordinate y3 or x3.
For n3 = 0 (vertical averaging along the axis of the cylinder), 2-Dimensional,
3-component solenoidal fields must be expanded along a complete basis for
fields derived from 2D stream functions:
curl(φne3), φne3
, φn = φn(r, θ),
−△φn = µnφn, φn|∂Γ = 0, and
curlΦn =
curl(φne3), µnφne3
a, be3
denotes a 3-component vector whose horizontal projection is
a and vertical projection is be3.
Let us explicit elementary swirling wave flows which are exact solutions to
(2.1) and (2.7):
Lemma 2.3 For every n = (n1, n2, n3), the following quasiperiodic (T
time) solenoidal fields are exact solution of the full 3D nonlinear Euler equa-
tions (2.1):
V(t, y) =
e3 × y + exp(
Jt)Φn(exp(−
Jt)y) exp(±i
αΩt), (2.16)
n3 is the vertical wave number of Φn and exp(
Jt) the unitary group of rigid
body rotations:
0 −1 0
1 0 0
0 0 0
 , eΩJt/2 =
cos(Ωt
) − sin(Ωt
sin(Ωt
) cos(Ωt
0 0 1
 . (2.17)
Remark 2.4 These fields are exact quasiperiodic, nonaxisymmetric swirling
flow solutions of the 3D Euler equations. For n3 6= 0, their second components
Ṽ(t, y) = exp(
Jt)Φn(exp(−
Jt)y) exp(± in3
αΩt) (2.18)
are Beltrami flows (curlṼ×Ṽ ≡ 0) exact solutions of (2.7) with Ṽ(t = 0, y) =
Φn(y).
Ṽ(t, y) in Eq. (2.18) are dispersive waves with frequencies Ω
and n3α|λn|Ω,
where α = 2π
. Moreover, each Ṽ(t, y) is a traveling wave along the cylinder’s
axis, since it contains the factor
iαn3(±z ±
Note that n3 large corresponds to small axial (vertical) scales, albeit 0 ≤
α|n3/λn| ≤ 1.
Proof of Lemma 2.3. Through the canonical rigid body transformation for
both the field V(t, y) and the space coordinates y = (y1, y2, y3):
V(t, y) = e+ΩJt/2U(t, e−ΩJt/2y) +
Jy, x = e−ΩJt/2y, (2.19)
the 3D Euler equations (2.1), (2.2) transform into:
∂tU+ (curlU+Ωe3)×U = −∇
(|x1|2 + |x2|2) +
, (2.20)
∇ ·U = 0, U(t, x)|t=0 = U(0) = Ṽ0(x), (2.21)
For Beltrami flows such that curlU×U ≡ 0, these Euler equations (2.20)-
(2.21) in a rotating frame reduce to:
∂tU+Ωe3 ×U+∇π = 0, ∇ ·U = 0,
which are identical to the Poincaré-Sobolev nonlocal wave equations in the
cylinder [M-N-B-G], [Poi], [Sob], [Ar-Khe]:
∂tΨ+Ωe3 ×Ψ+∇π = 0, ∇ ·Ψ = 0, (2.22)
curl2Ψ−Ω2 ∂
Ψ = 0, Ψ ·N|∂D = 0. (2.23)
It suffices to verify that the Beltrami flows Ψn(t, x) = Φn(x) exp
±iαn3|λn|Ωt
where Φ±n (x) and ±|λn| are curl eigenfunctions and eigenvalues, are exact
solutions to the Poincaré-Sobolev wave equation, in such a rotating frame of
reference.
Remark 2.5 The frequency spectrum of the Poincaré vorticity waves (solu-
tions to (2.22)) is exactly ±iαn3|λn|Ω, n = (n1, n2, n3) indexing the spectrum
of curl. Note that n3 = 0 (zero frequency of rotating waves) corresponds to
2-Dimensional, 3-Components solenoidal vector fields.
We now transform the Cauchy problem for the 3D Euler equations (2.1)-
(2.2) into an infinite dimensional nonlinear dynamical system by expanding
V(t, y) along the swirling wave flows (2.16)-(2.18):
V(t, y) =
e3 × y
(2.24a)
+ exp
n=(n1,n2,n3)
un(t) exp
(2.24b)
V(t = 0, y) =
e3 × y + Ṽ0(y)
(2.24c)
Ṽ0(y) =
n=(n1,n2,n3)
un(0)Φn(y),
(2.24d)
where Φn denotes the curl eigenfunctions of Proposition 2.1 if n3 6= 0, and
curl(φne3), φne3
if n3 = 0 (2D case, Remark 2.2).
As we focus on the case where helicity is conserved for (2.1)-(2.2), we
consider the class of initial data Ṽ0 such that [M-N-B-G]:
curlṼ0 ·N = 0 on Γ,
where Γ is the lateral boundary of the cylinder.
The infinite dimensional dynamical system is then equivalent to the 3D
Euler equations (2.1)-(2.2) in the cylinder, with n = (n1, n2, n3) ranging over
the whole spectrum of curl, e.g.:
k3+m3=n3
k2+m2=n2
× < curlΦk ×Φm,Φn > uk(t)um(t)
(2.25)
curlΦ±k = ±λkΦ
k if k3 6= 0,
curlΦk =
curl(φke3), µkφke3
if k3 = 0
(2D, 3-components, Remark 2.2), similarly for m3 = 0 and n3 = 0. The inner
product < , > denotes the L2 complex-valued inner product in D.
This is an infinite dimensional system of coupled equations with quadratic
nonlinearities, which conserve both the energy
E(t) =
|un(t)|2
and the helicity
Hel(t) =
±|λn| |u±n (t)|2.
The quadratic nonlinearities split into resonant terms where the exponential
oscillating phase factor in (2.25) reduces to unity and fast oscillating non-
resonant terms (Ω >> 1). The resonant set K is defined in terms of vertical
wavenumbers k3,m3, n3 and eigenvalues ±λk, ±λm, ±λn of curl:
K = {± k3
= 0, n3 = k3 +m3, n2 = k2 +m2}. (2.27)
Here k2,m2, n2 are azimuthal wavenumbers.
We shall call the “resonant Euler equations” the following ∞-dimensional
dynamical system restricted to (k,m, n) ∈ K:
(k,m,n)∈K
< curlΦk ×Φm,Φn > ukum = 0, (2.28a)
un(0) ≡< Ṽ0,Φn >, (2.28b)
here curlΦ±k = ±λkΦ
k if k3 6= 0, curlΦk =
curl(φke3), µkφke3
if k3 = 0;
similarly for m3 = 0 and n3 = 0 (2D components, Remark 2.2). If there
are no terms in (2.28a) satisfying the resonance conditions, then there will be
some modes for which
Lemma 2.6 The resonant 3D Euler equations (2.28) conserve both energy
E(t) and helicity Hel(t). The energy and helicity are identical to that of the
full exact 3D Euler equations (2.1)-(2.2).
The set of resonances K is studied in depth in [M-N-B-G]. To summarize,
K splits into:
(i ) 0-wave resonances, with n3 = k3 = m3 = 0; the corresponding reso-
nant equations are identical to the 2-Dimensional, 3-Components Euler
equations, with I.C.
Ṽ0(y1, y2, y3) dy3.
(ii) Two-Wave resonances, with k3m3n3 = 0, but two of them are not null;
the corresponding resonant equations (called “catalytic equations”) are
proven to possess an infinite, countable set of new conservation laws
[M-N-B-G].
(iii) Strict three-wave resonances for a subset K∗ ⊂ K.
Definition 2.7 The set K∗ of strict 3 wave resonances is:
= 0, k3m3n3 6= 0, n3 = k3 +m3, n2 = k2 +m2
(2.29)
Note that K∗ is parameterized by h/R, since α = 2π
parameterizes the eigen-
values λn, λk, λm of the curl operator.
Proposition 2.8 There exist a countable, non-empty set of parameters h
which K∗ 6= ∅.
Proof. The technical details, together with a more precise statement, are
postponed to the proof of Lemma 3.7. Concrete examples of resonant ax-
isymmetric and helical waves are discussed in [Mah] ( cf. Figure 2 in the
article).
Corollary 2.9 Let
Ṽ0(y1, y2, y3) dy3 = 0, i.e. zero vertical mean for the
I.C. Ṽ0(y) in (2.2), (2.8), (2.24d) and (2.28b). Then the resonant 3D Euler
equations are invariant on K∗:
(k,m,n)∈K∗
λk < Φk ×Φm,Φn > ukum = 0, k3m3n3 6= 0, (2.30a)
un(0) =< Ṽ0,Φn > (2.30b)
(where Ṽ0 has spectrum restricted to n3 6= 0).
Proof. This is an immediate corollary of the “operator splitting” Theorem
3.2 in [M-N-B-G]. �
We shall call the above dynamical systems the “strictly resonant Euler
system”. This is an ∞-dimensional Riccati system which conserves Energy
and Helicity. It corresponds to nonlinear interactions depleted on K∗.
How do dynamics of the resonant Euler equations (2.28) or (2.30) approx-
imate exact solutions of the Cauchy problem for the full Euler equations in
strong norms? This is answered by the following theorem, proven in Section
4 of [M-N-B-G]:
Theorem 2.10 Consider the initial value problem
V(t = 0, y) =
e3 × y + Ṽ0(y), Ṽ0 ∈ Hsσ, s > 7/2
for the full 3D Euler equations, with ||Ṽ0||Hs
≤M0s and curlṼ0 ·N = 0 on Γ.
• Let V(t, y) = Ω
e3 × y + Ṽ(t, y) denote the solution to the exact Euler
equations.
• Let w(t, x) denote the solution to the resonant 3D Euler equations with
Initial Condition w(0, x) ≡ w(0, y) = Ṽ0(y).
• Let ||w(t, y)||Hsσ ≤Ms(TM ,M
s ) on 0 ≤ t ≤ TM , s > 7/2.
Then, ∀ǫ > 0, ∃ Ω∗(TM ,M0s , ǫ) such that, ∀Ω ≥ Ω∗:
Ṽ(t, y)− exp
un(t)e
−i n3
on 0 ≤ t ≤ TM , ∀β ≥ 1, β ≤ s− 2. Here || · ||Hβ is defined in (1.13).
The 3D Euler flow preserves the condition curlṼ0 · N = 0 on Γ, that is
curlV(t, y) · N = 0 on Γ, for every t ≥ 0 [M-N-B-G]. The proof of this
“error-shadowing” theorem is delicate, beyond the usual Gronwall differential
inequalities and involves estimates of oscillating integrals of almost periodic
functions of time with values in Banach spaces. Its importance lies in that
solutions of the resonant Euler equations (2.28) and/or (2.30) are uniformly
close in strong norms to those of the exact Euler equations (2.1)-(2.2), on
any time interval of existence of smooth solutions of the resonant system.
The infinite dimensional Riccati systems (2.28) and (2.30) are not just hydro-
dynamic models, but exact asymptotic limit systems for Ω ≫ 1. This is in
contrast to all previous literature on conservative 3D hydrodynamic models,
such as in [G-D-O].
3 Strictly resonant Euler systems: the SO(3)
We investigate the structure and the dynamics of the “strictly resonant Euler
systems” (2.30). Recall that the set of 3-wave resonances is:
(k,m, n) : ± k3
= 0, k3m3n3 6= 0,
n3 = k3 +m3, n2 = k2 +m2
(3.1)
From the symmetries of the curl eigenfunctions Φn and eigenvalues λn in the
cylinder, the following identities hold under the transformation n2 → −n2,
n3 → −n3
Φ(n1,−n2,−n3) = Φ∗(n1, n2, n3) ,
λ(n1,−n2,−n3) = λ(n1, n2, n3) .
(3.2)
where ∗ designates the complex conjugate (see Section 3, [M-N-B-G] for
details). The eigenfunctions Φ(n1, n2, n3) involve the radial functions
Jn2(β(n1, n2, αn3)r) and J
(β(n1, n2, αn3)r), with
λ2(n1, n2, n3) = β
2(n1, n2, αn3) + α
2n23;
β(n1, n2, αn3) are discrete, countable roots of equation (3.30) in [M-N-B-G],
obtained via an equivalent Sturm-Liouville radial problem. Since the curl
eigenfunctions are even in r → −r, n1 → −n1, we will extend the indices
n1 = 1, 2, ...,+∞ to −n1 = −1,−2, ... with the above radial symmetry in
mind.
Corollary 3.1 The 3-wave resonance set K∗ is invariant under the symme-
tries σj , j = 0, 1, 2, 3, where
σ0(n1, n2, n3) = (n1, n2, n3),
σ1(n1, n2, n3) = (−n1, n2, n3),
σ2(n1, n2, n3) = (n1,−n2, n3)
σ3(n1, n2, n3) = (n1, n2,−n3) .
Remark 3.2 For 0 < i ≤ 3, 0 < j ≤ 3, 0 < l ≤ 3 σ2j = Id, σiσj = −σl if
i 6= j and σiσjσl = −Id, for i 6= j 6= l. The σj do preserve the convolution
conditions in K∗.
We choose an α for which the set K∗ is not empty. We further take the
hypothesis of a single triple wave resonance (k,m, n), modulo the symmetries
Hypothesis 3.3 K∗ is such that there exists a single triple wave number
resonance (n, k,m), modulo the symmetries σj , j = 1, 2, 3 and σj(k) 6=
k, σj(m) 6= m, σj(n) 6= n for j = 2 and j = 3.
Under the above hypothesis, one can demonstrate that the strictly resonant
Euler system splits into three uncoupled systems in C3:
Theorem 3.4 Under hypothesis 3.3, the resonant Euler system reduces to
three uncoupled rigid body systems in C3:
+ i(λk − λm)CkmnUkUm = 0 (3.3a)
− i(λm − λn)CkmnUnU∗m = 0 (3.3b)
− i(λn − λk)CkmnUnU∗k = 0 (3.3c)
where Ckmn = i < Φk ×Φm,Φ∗n >, Ckmn real and the other two uncoupled
systems obtained with the symmetries σ2(k,m, n) and σ3(k,m, n). The energy
and the helicity of each subsystem are conserved:
k + UmU
m + UnU
n) = 0,
(λkUkU
k + λmUmU
m + λnUnU
n) = 0.
Proof. It follows from U−k = U
k , λ(−k) = λ(+k), similarly for m and
n; and in a very essential way from the antisymmetry of < Φk ×Φm,Φ∗n >,
together with curlΦk = λkΦk. That Ckmn is real follows from the eigenfunc-
tions explicited in Section 3 of [M-N-B-G]. �
Remark 3.5 This deep structure, i.e. SO(3;C) rigid body systems in C3 is
a direct consequence of the Lamé form of the full 3D Euler equations, cf. Eqs.
(1.3) and (2.7), and the nonlinearity curlV ×V.
The system (3.3) is equivariant with respect to the symmetry operators
(z1, z2, z3) → (z∗1 , z∗2 , z∗3), (z1, z2, z3) → (exp(iχ1)z1, exp(iχ2)z2, exp(iχ3)z3) ,
provided χ1 = χ2 + χ3. It admits other integrals known as the Manley-
Rowe relations (see, for instance [We-Wil]). It differs from the usual 3-
wave resonance systems investigated in the literature, such as in [Zak-Man1],
[Zak-Man2], [Gu-Ma] in that
(1) helicity is conserved,
(2) dynamics of these resonant systems rigorously “shadow” those of the
exact 3D Euler equations, see Theorem 2.10.
Real forms of the system (3.3) are found in Gledzer et al. [G-D-O], corre-
sponding to the exact invariant manifold Uk ∈ iR, Um ∈ R, Un ∈ R, albeit
without any rigorous asymptotic justification. The C3 systems (3.3) with
helicity conservation laws are not discussed in [G-D-O].
The only nontrivial Manley-Rowe conservation laws for the resonant sys-
tem (3.3), rigid body SO(3;C), which are independent from energy and he-
licity, are:
(rkrmrn sin(θn − θk − θm)) = 0,
where Uj = rj exp(iθj), j = k,m, n, and
E1 = (λk − λm)r2n − (λm − λn)r2k,
E2 = (λm − λn)r2k − (λn − λk)r2m.
The resonant system (3.3) is well known to possess hyperbolic equilibria
and heteroclinic/homoclinic orbits on the energy surface. We are interested
in rigorously proving arbitrary large bursts of enstrophy and higher norms
on arbitrarily small time intervals, for properly chosen h/R. To simplify
the presentation, we establish the results for the simpler invariant manifold
Uk ∈ iR, and Um, Un ∈ R.
Rescale time as:
t→ t/Ckmn.
Start from the system
U̇n + i(λk − λm)UkUm = 0
U̇k − i(λm − λn)UnU∗m = 0
U̇m − i(λn − λk)UnU∗k = 0
(3.4)
Assume that Uk ∈ iR and that Um, Un ∈ R: set p = iUk, q = Um and r = Un,
as well as λk = λ, λm = µ and λn = ν: then
ṗ+ (µ− ν)qr = 0
q̇ + (ν − λ)rp = 0
ṙ + (λ− µ)pq = 0
(3.5)
This system admits two first integrals:
E = p2 + q2 + r2 (energy)
H = λp2 + µq2 + νr2 (helicity)
(3.6)
System (3.5) is exactly the SO(3,R) rigid body dynamics Euler equations,
with inertia momenta Ij =
|λj | , j = k,m, n [Ar1].
Lemma 3.6 ([Ar1], [G-D-O]) With the ordering λk > λm > λn, i.e. λ > µ >
ν, the equilibria (0,±1, 0) are hyperbolic saddles on the unit energy sphere,
and the equilibria (±1, 0, 0), (0, 0,±1) are centers. There exist equivariant
families of heteroclinic connections between (0,+1, 0) and (0,−1, 0). Each
pair of such connections correspond to equivariant homoclinic cycles at (0, 1, 0)
and (0,−1, 0).
We investigate bursting dynamics along orbits with large periods, with
initial conditions close to the hyperbolic point (0, E(0), 0) on the energy sphere
E. We choose resonant triads such that λk > 0, λn < 0, λk ∼ |λn|, |λm| ≪ λk,
equivalently:
λ > µ > ν, λν < 0, |µ| ≪ λ and λ ∼ |ν|. (3.7)
Lemma 3.7 There exist h/R with K∗ 6= ∅, such that
λk > λm > λn, λkλn < 0, |λm| ≪ λk and λk ∼ |λn|.
Remark 3.8 Together with the polarity ± of the curl eigenvalues, these are
3-wave resonances where two of the eigenvalues are much larger in mod-
uli than the third one. In the limit |k|, |m|, |n| ≫ 1, λk ∼ ±|k|, λm ∼
±|m|, λn ∼ ±|n|, the eigenfunctions Φ have leading asymptotic terms which
involve cosines and sines periodic in r, cf. Section 3 [M-N-B-G]. In the
strictly resonant equations (2.30), the summation over the quadratic terms
becomes an asymptotic convolution in n1 = k1+n1. The resonant three waves
in Lemma 3.7 are equivalent to Fourier triads k + m = n, with |k| ∼ |n|
and |m| ≪ |k|, |n|, in periodic lattices. In the physics of spectral theory of
turbulence [Fri], [Les], these are exactly the triads responsible from transfer of
energy between large scales and small scales. These are the triads which have
hampered mathematical efforts at proving the global regularity of the Cauchy
problem for 3D Navier-Stokes equations in periodic lattices [Fe].
Proof of Lemma 3.7 ([M-N-B-G]) The transcendental dispersion law for
3-waves in K∗ for cylindrical domains, is a polynomial of degree four in ϑ3 =
1/h2:
P̃ (ϑ3) = P̃4ϑ
3 + P̃3ϑ
3 + P̃2ϑ
3 + P̃1ϑ3 + P̃0 = 0, (3.8)
with n2 = k2 +m2 and n3 = k3 +m3.
Then with hk =
β2(k1,k2,αk3)
, hm =
β2(m1,m2,αm3)
, hn =
β2(n1,n2,αn3)
, cf.
the radial Sturm-Liouville problem in Section 3, [M-N-B-G], the coefficients
of P̃ (ϑ3) are given by:
P̃4 = −3,
P̃3 = −4(hk + hm + hn),
P̃2 = −6(hkhm + hkhn + hmhn),
P̃1 = −12hkhmhn,
P̃0 = h
n + h
n + h
k − 2(hkhmh2n + hkhnh2m + hmhnh2k).
Similar formulas for the periodic lattice domain were first derived in [B-M-N2],
[B-M-N3], [B-M-N4]. In cylindrical domains the resonance condition for K∗
is identical to
ϑ3 + hk
ϑ3 + hm
ϑ3 + hn
with ϑ3 =
, hk = β
2(k)/k23 , hm = β
2(m)/m23, hn = β
2(n)/n23; Eq. (3.8) is
the equivalent rational form.
From the asymptotic formula (3.44) in [M-N-B-G], for large β:
β(n1, n2, n3) ∼ n1π + n2
+ ψ, (3.9)
where ψ = 0 if lim m2
= 0 (e.g. h fixed, m2/m3 → 0) and ψ = ±π2 if
lim m2
= ±∞ (e.g. m2
fixed, h → ∞). The proof is completed by taking
leading terms P̃0+ϑ3P̃1 in (3.8), ϑ3 =
≪ 1, and m2 = 0, k2 = O(1), n2 =
O(1). �
We now state a theorem for bursting of the H3 norm in arbitrarily small
times, for initial data close to the hyperbolic point (0, E(0), 0):
Theorem 3.9 (Bursting dynamics in H3). Let λ > µ > ν, λν < 0, |µ| ≪ λ
and λ ∼ |ν|. Let W (t) = λ6p(t)2 + µ6q(t)2 + ν6r(t)2 the H3-norm squared of
an orbit of (3.5). Choose initial data such that: W (0) = λ6p(0)2 + µ6q(0)2
with λ6p(0)2 ∼ 1
W (0) and µ6q(0)2 ∼ 1
W (0). Then there exists t∗ > 0, such
W (t) ≥
W (0)
where t∗ ≤ 6√
W (0)
µ2Ln(λ/|µ|)(λ/|µ|)−1.
Remark 3.10 Under the conditions of Lemma 3.7,
≫ 1, whereas
µ2(Ln(λ/|µ|))(λ/|µ|)−1 ≪ 1. Therefore, over a small time interval of length
O(µ2(Ln(λ/|µ|))(λ/|µ|)−1) ≪ 1, the ratio ||U(t)||H3/||U(0)||H3 grows up to
a maximal value O
(λ/|µ|)3
≫ 1. Since the orbit is periodic, the H3 semi-
norm eventually relaxes to its initial state after some time (this being a mani-
festation of the time-reversibility of the Euler flow on the energy sphere). The
“shadowing” theorem 2.10 with s > 7/2 ensures that the full, original 3D Eu-
ler dynamics, with the same initial conditions, will undergo the same type of
burst. Notice that, with the definition (1.13) of ‖ · ‖Hs , one has
||Ωe3 × y||H3 = ||curl3(Ωe3 × y)||L2 = 0 .
Hence the solid rotation part of the original 3D Euler solution does not con-
tribute to the ratio ||V(t)||H3/||V(0)||H3 .
Theorem 3.11 (Bursting dynamics of the enstrophy). Under the same con-
ditions for the 3-wave resonance, let Ξ(t) = λ2p(t)2 + µ2q(t)2 + ν2r(t)2 the
enstrophy. Choose initial data such that Ξ(0) = λ2p(0)2 + µ2q(0)2 + ν2r(0)2
with λ2p(0)2 ∼ 1
Ξ(0), µ2q(0)2 ∼ 1
Ξ(0). Then there exists t∗∗ > 0, such that
Ξ(t∗∗) ≥
where t∗∗ ≤ 1√
Ln (λ/|µ|) (λ/|µ|)−1 .
Remark 3.12 It is interesting to compare this mechanism for bursts with ear-
lier results in the same direction obtained by DiPerna and Lions. Indeed, for
each p ∈ (1,∞), each δ ∈ (0, 1) and each t > 0, Di Perna and Lions [DiPe-Li]
constructed examples of 2D-3 components solutions to Euler equations such
||V(0)||W 1,p ≤ ǫ while ||V(t)||W 1,p ≥ 1/δ .
Their examples essentially correspond to shear flows of the form
V(t, x1, x2) =
u(x2)
w(x1 − tu(x2), x2)
where u ∈W 1,px2 while w ∈W
. Obviously
curlV(t, x1, x2) =
(∂2 − tu′(x2)∂1)w(x1 − tu(x2), x2)
−∂1w(x1 − tu(x2), x2)
−u′(x2)
Thus, all components in curlV(t, x1, x2) belong to L
loc, except for the term
−tu′(x2)∂1w(x1 − tu(x2), x2) .
For each t > 0, this term belongs to Lp for all choices of the functions u ∈
W 1,px2 and w ∈ W
x1,x2
if and only if p = ∞. Whenever p < ∞, DiPerna and
Lions construct their examples as some smooth approximation of the situation
above in the strong W 1,p topology.
In other words, the DiPerna-Lions construction works only in cases where
the initial vorticity does not belong to an algebra — specifically to Lp, which
is not an algebra unless p = ∞.
The type of burst obtained in our construction above is different: in that
case, the original vorticity belongs to the Sobolev space H2, which is an algebra
in space dimension 3. Similar phenomena are observed in all Sobolev spaces
Hβ with β ≥ 2 — which are also algebras in space dimension 3.
In other words, our results complement those of DiPerna-Lions on bursts
in higher order Sobolev spaces, however at the expense of using more intricate
dynamics.
We proceed to the proofs of Theorem 3.9 and 3.11. We are interested in
the evolution of
Ξ = λ2p2 + µ2q2 + ν2r2 (enstrophy) (3.10)
Compute
Ξ̇ = −2
λ2(µ− ν) + µ2(ν − λ) + ν2(λ− µ)
pqr (3.11)
˙(pqr) = −(µ− ν)q2r2 − (ν − λ)r2p2 − (λ− µ)p2q2 (3.12)
Using the first integrals above, one has
 (3.13)
where V an is the Vandermonde matrix
V an =
1 1 1
λ µ ν
λ2 µ2 ν2
For λ 6= µ 6= ν 6= λ, this matrix is invertible and
V an−1 =
(λ−µ)(λ−ν)
−(µ+ν)
(λ−µ)(λ−ν)
(λ−µ)(λ−ν)
(µ−ν)(µ−λ)
−(ν+λ)
(µ−ν)(µ−λ)
(µ−ν)(µ−λ)
(ν−λ)(ν−µ)
−(λ+µ)
(ν−λ)(ν−µ)
(ν−λ)(ν−µ)
Hence
(λ− µ)(λ − ν)
(Ξ− (µ+ ν)H + µνE)
(µ− ν)(µ− λ)
(Ξ− (ν + λ)H + νλE)
(ν − λ)(ν − µ)
(Ξ− (λ+ µ)H + λµE)
(3.14)
so that
(µ− ν)q2r2 = − (Ξ− (ν + λ)H + νλE) (Ξ− (λ + µ)H + λµE)
(λ− µ)(λ − ν)(µ− ν)
(ν − λ)r2p2 = − (Ξ− (λ + µ)H + λµE) (Ξ− (µ+ ν)H + µνE)
(λ− µ)(λ − ν)(µ− ν)
(λ− µ)p2q2 = − (Ξ− (µ+ ν)H + µνE) (Ξ− (ν + λ)H + νλE)
(λ− µ)(λ − ν)(µ− ν)
Later on, we shall use the notations
x−(λ, µ, ν) = (µ+ ν)H − µνE
x0 (λ, µ, ν) = (µ+ λ)H − µλE
x+(λ, µ, ν) = (λ+ ν)H − λνE
(3.15)
Therefore, we find that Ξ satisfies the second order ODE
Ξ̈ =− 2Kλ,µ,ν ((Ξ− x−(λ, µ, ν))(Ξ − x0(λ, µ, ν))
+(Ξ− x0(λ, µ, ν))(Ξ − x+(λ, µ, ν)) + (Ξ− x+(λ, µ, ν))(Ξ − x0(λ, µ, ν)))
which can be put in the form
Ξ̈ = −2Kλ,µ,νP ′λ,µ,ν(Ξ) (3.16)
where Pλ,µ,ν is the cubic
Pλ,µ,ν(X) = (X − x−(λ, µ, ν))(X − x0(λ, µ, ν))(X − x+(λ, µ, ν)) (3.17)
Kλ,µ,ν =
λ2(µ− ν) + µ2(ν − λ) + ν2(λ− µ)
(λ− µ)(λ− ν)(µ − ν)
(3.18)
In the sequel, we assume that the initial data for (p, q, r) is such that
r(0) = 0 , p(0)(q(0) 6= 0
Let us compute
x−(λ, µ, ν) = λνp(0)
2 + µ2q(0)2 + µ(λ− ν)p(0)2
x0 (λ, µ, ν) = λ
2p(0)2 + µ2q(0)2
x+(λ, µ, ν) = λ
2p(0)2 +
ν + λ
µ2q(0)2
(3.19)
We shall also assume that
λ > µ > ν , λν < 0 , |µ| ≪ λ and λ ∼ |ν| (3.20)
Then Kλ,µ,ν > 0 — in fact Kλ,µ,ν ∼ 2, and Ξ is a periodic function of t such
Ξ(t) = x0(λ, µ, ν) , sup
Ξ(t) = x+(λ, µ, ν) (3.21)
with half-period
Tλ,µ,ν =
Kλ,µ,ν
∫ x+(λ,µ,ν)
x0(λ,µ,ν)
−Pλ,µ,ν(x)
(3.22)
We are interested in the growth of the (squared) H3 norm
W (t) = λ6p(t)2 + µ6q(t)2 + ν6r(t)2 (3.23)
Expressing p2, q2 and r2 in terms of E, H and Ξ, it is found that
λ6(Ξ− x−(λ, µ, ν))
(λ− µ)(λ − ν)
µ6(Ξ− x+(λ, µ, ν))
(µ− ν)(µ− λ)
ν6(Ξ− x0(λ, µ, ν))
(ν − λ)(ν − µ)
(3.24)
Hence, when Ξ = x+(λ, µ, ν), then
λ6(x+(λ, µ ν) − x−(λ, µ, ν))
(λ− µ)(λ− ν)
ν6(x+(λ, µ ν)− x0(λ, µ, ν))
(ν − λ)(ν − µ)
λ6(x+(λ, µ ν) − x−(λ, µ, ν))
(λ− µ)(λ− ν)
Let us compute
x+(λ, µ ν)− x−(λ, µ, ν) = (λ − µ)(λ− ν)p(0)2 +
ν + λ
µ2q(0)2
& −νλq(0)2 ∼ λ2q(0)2
(3.25)
We shall pick the initial data such that
W (0) = λ6p(0)6 + µ6q(0)6 with λ6p(0)2 ∼ 1
W (0) and µ6q(0)2 ∼ 1
W (0)
(3.26)
Hence, when Ξ reaches x+(λ, µ, ν), one has
λ8q(0)2
(λ− µ)(λ− ν)
µ6(λ − µ)(λ− ν)
W (0) ∼ 1
W (0) . (3.27)
Hence W jumps from W (0) to a quantity ∼ 1
W (0) in an interval of time
that does not exceed one period of the Ξ motion, i.e. 2Tλ,µ,ν . Let us estimate
this interval of time. We recall the asymptotic equivalent for the period of an
elliptic integral in the modulus 1 limit.
Lemma 3.13 Assume that x− < x0 < x+. Then
(x− x−)(x− x0)(x+ − x)
x+ − x−
x+−x0
x+−x−
uniformly in x−, x0, and x+ as
x+−x0
x+−x− → 1.
x+(λ, µ, ν) − x−(λ, µ, ν)
λ2q(0)2
∼ |µ|
W (0)
x0(λ, µ, ν) − x−(λ, µ, ν) = (λ− µ)(λ − ν)p(0)2 (3.28)
so that
x+−x0
x+−x−
1− (λ−µ)(λ−ν)p(0)
(λ−µ)(λ−ν)p(0)2+(µ(ν+λ)−νλ−µ2)q(0)2
∼ (λ− µ)(λ− ν)p(0)
2 + (µ(ν + λ)− νλ− µ2)q(0)2
2(λ− µ)(λ − ν)p(0)2
∼ q(0)
2p(0)2
W (0)/2µ6
W (0)/2λ6
Hence
2Tλ,µ,ν .
W (0)
≤ 12√
W (0)
(3.29)
Conclusion: collecting (3.26), (3.27) and (3.29), we see that the squared H3
norm W varies from W (0) to a quantity ∼ ρ6W (0) in an interval of time
. 12√
W (0)
µ2 ln ρ
. (Here ρ = λ/µ).
We now proceed to obtain similar bursting estimates for the enstrophy.
We return to (3.21) and (3.22). Pick the initial data so that
Ξ(0) = λ2p(0)2 + µ2q(0)2 with λ2p(0)2 ∼ 1
Ξ(0) and µ2q(0)2 ∼ 1
Ξ(0).
x+(λ, µ, ν) − x−(λ, µ, ν)
= (λ− µ)(λ − ν)p(0)2 +
ν + λ
µ2q(0)2
∼ 2λ2p(0)2 + λ2q(0)2 ∼
while
x0(λ, µ, ν)− x−(λ, µ, ν) = (λ− µ)(λ− ν)p(0)2 ∼ 2λ2p(0)2 ∼ Ξ(0).
Hence, in the limit as ρ = λ/|µ| → +∞, one has
2Tλ,µ,ν ∼
ρ2Ξ(0)
1− Ξ(0)1
ρ2Ξ(0)
2Ξ(0)
1− 2ρ−2
2Ξ(0)
And Ξ varies from
x0(λ, µ, ν) = Ξ(0) to x+(λ, µ, ν) ∼ ρ2Ξ(0)
on an interval of time of length Tλ,µ,ν . �
4 Strictly resonant Euler systems: the case of
3-waves resonances on small-scales
4.1 Infinite dimensional uncoupled SO(3) systems
In this section, we consider the 3-wave resonant set K∗ when
|k|2, |m|2, |n|2 ≥
, 0 < η ≪ 1,
i.e. 3-wave resonances on small scales; here |k|2 = k21 + k22 + k23 , where
(k1, k2, k3) index the curl eigenvalues, and similarly for |m|2, |n|2. Recall
that k2 + m2 = n2, k3 + m3 = n3 (exact convolutions), but that the sum-
mation on k1, m1 on the right hand side of Eqs. (2.30) is not a convolution.
However, for |k|2, |m|2, |n|2 ≥ 1
, the summation in k1, m1 becomes an
asymptotic convolution. First:
Proposition 4.1 The set K∗ restricted to |k|2, |m|2, |n|2 ≥ 1
, ∀η, 0 < η ≪ 1
is not empty: there exist at least one h/R with resonant three waves satisfying
the above small scales condition.
Proof. We follow the algebra of the exact transcendental dispersion law
(3.8) derived in the proof of Lemma 3.7. Note that P̃ (ϑ3) < 0 for ϑ3 =
large enough. We can choose hm =
β2(m1,m2,αm3)
= 0, say in the specific limit
→ 0, and β(m1,m2, αm3) ∼ m1π+m2 π2 +
. Then P̃0 = h
n > 0 and
P̃ (ϑ3) must possess at least one (transcendental) root ϑ3 =
In the above context, the radial components of the curl eigenfunctions in-
volve cosines and sines in βr
(cf. Section 3, [M-N-B-G]) and the summation in
k1, m1 on the right hand side of the resonant Euler equations (2.30) becomes
an asymptotic convolution. The rigorous asymptotic convolution estimates are
highly technical and detailed in [Fro-M-N]. The 3-wave resonant systems for
|k|2, |m|2, |n|2 ≥ 1
are equivalent to those of an equivalent periodic lattice
[0, 2π]× [0, 2π]× [0, 2πh], ϑ3 = 1h2 ; the resonant three wave relation becomes:
ϑ3 + ϑ1
ϑ3 + ϑ1
ϑ3 + ϑ1
= 0, (4.1a)
k +m = n, k3m3n3 6= 0. (4.1b)
The algebraic geometry of these rational 3-wave resonance equations has been
investigated in depth in [B-M-N3] and [B-M-N4]. Here ϑ1, ϑ2, ϑ3 are periodic
lattice parameters; in the small-scales cylindrical case, ϑ1 = ϑ2 = 1 (after
rescaling of n2, k2, m2), ϑ3 = 1/h
2, h height. Based on the algebraic
geometry of “resonance curves” in [B-M-N3], [B-M-N4], we investigate the
resonant 3D Euler equations (2.30) in the equivalent periodic lattices.
First, triplets (k,m, n) solution of (4.1) are invariant under the reflec-
tion symmetries σ0, σ1, σ2, σ3 defined in Corollary 3.1 and Remark 3.2: σ0 =
Id, σj(k) = (ǫi,jki), 1 ≤ i ≤ 3, ǫi,j = +1 if i 6= j, ǫi,j = −1 if i = j, 1 ≤ j ≤ 3.
Second the set K∗ in (4.1) is invariant under the homothetic transformations:
(k,m, n) → (γk, γm, γn), γ rational. (4.2)
The resonant triplets lie on projective lines in the wavenumber space, with
equivariance under σj , 0 ≤ j ≤ 3 and γ-rescaling. For every given equivariant
family of such projective lines, the resonant curve is the graph of ϑ3
versus
, for parametric domain resonances in ϑ1, ϑ2, ϑ3.
Lemma 4.2 (p.17, [B-M-N4]). For every equivariant (k,m, n), the resonant
curve in the quadrant ϑ1 > 0, ϑ2 > 0, ϑ3 > 0 is the graph of a smooth function
ϑ3/ϑ1 ≡ F (ϑ2/ϑ1) intersected with the quadrant.
Theorem 4.3 (p.19, [B-M-N4]). A resonant curve in the quadrant ϑ3/ϑ1 > 0,
ϑ2/ϑ1 > 0 is called irreducible if:
k23 k
m23 m
n23 n
 6= 0. (4.3)
An irreducible resonant curve is uniquely characterized by six non-negative
algebraic invariants P1, P2, R1, R2, S1, S2, such that
P21 ,P22
R21,R22
S21 ,S22
and permutations thereof.
Lemma 4.4 (p. 25, [B-M-N4]). For resonant triplets (k,m, n) associated to
a given irreducible resonant curve, that is verifying Eq. (4.3), consider the
convolution equation n = k +m. Let σi(n) 6= n, ∀i, 1 ≤ i ≤ 3. Then there
are no more that two solutions (k,m) and (m, k), for a given n, provided
the six non-degeneracy conditions (3.39)-(3.44) in [B-M-N4] for the algebraic
invariants of the irreducible curve are verified.
For more details on the technical non-degeneracy conditions, see the Ap-
pendix. An exhaustive algebraic geometric investigation of all solutions to
n = k +m on irreducible resonant curves is found in [B-M-N4]. The essence
of the above lemma lies in that given such an irreducible, “non-degenerate”
triplet (k,m, n) on K∗, all other triplets on the same irreducible resonant
curves are exhaustively given by the equivariant projective lines:
(k,m, n) → (γk, γm, γn), for some γ rational , (4.4)
(k,m, n) → (σjk, σjm,σjn), j = 1, 2, 3, (4.5)
and permutations of k and m in the above. Of course the homothety γ and
the σj symmetries preserve the convolution. This context of irreducible, “non-
degenerate” resonant curves yields an infinite dimensional, uncoupled system
of rigid body SO(3;R) and SO(3;C) dynamics for the 3D resonant Euler
equations (2.30).
Theorem 4.5 For any irreducible triplet (k,m, n) which satisfy Theorem 4.3,
and under the “non-degeneracy” conditions of Lemma 4.4 (cf. Appendix), the
resonant Euler equations split into the infinite, countable sequence of uncou-
pled SO(3;R) systems:
ȧk = Γkmn(λm − λn)aman, (4.6a)
ȧm = Γkmn(λn − λk)anak, (4.6b)
ȧn = Γkmn(λk − λm)akam, (4.6c)
for all (k,m, n) = γ(σj(k
∗), σj(m
∗), σj(n
∗)), γ = ±1,±2,±3..., 0 ≤ j ≤ 3.
(4.7)
k∗,m∗, n∗ are some relatively prime integer vectors in Z3 characterizing the
equivariant family of projective lines (k,m, n); Γkmn = i < Φk ×Φm,Φ∗n >,
Γkmn real.
Proof. Theorem 4.5 is a simpler version for invariant manifolds of more
general SO(3;C) systems. It is a straightforward corollary of Proposition 3.2,
Proposition 3.3, Theorem 3.3, Theorem 3.4 and Theorem 3.5 in [B-M-N4].
The latter article did not explicit the resonant equations and did not use the
curl-helicity algebra fundamentally underlying this present work. Rigorously
asymptotic infinite countable sequences of uncoupled SO(3;R), SO(3;C) sys-
tems are not derived via the usual harmonic analysis tools of Fourier modes,
in the 3D Euler context. Polarization of curl eigenvalues and eigenfunctions
and helicity play an essential role.
Corollary 4.6 Under the conditions λn∗ − λk∗ > 0, λk∗ − λm∗ > 0, the
resonant Euler systems (4.6) admit a disjoint, countable family of homoclinic
cycles. Moreover, under the conditions λn∗ ≫ +1, λm∗ ≪ −1, |λk∗ | ≪ λn∗ ,
each subsystem (4.6) possesses orbits whose Hs norms, s ≥ 1, burst arbitrarily
large in arbitrarily small times.
Remark 4.7 One can prove that there exists some Γmax, 0 < Γmax < ∞,
such that |Γkmn| < Γmax, for all (k,m, n) on the equivariant projective lines
defined by (4.7). Systems (4.6) “freeze” cascades of energy; their total enstro-
phy Ξ(t) =
(k,m,n)(λ
k(t) + λ
m(t) + λ
n(t)) remains bounded, albeit
with large bursts of Ξ(t)/Ξ(0), on the reversible orbits topologically close to
the homoclinic cycles.
4.2 Coupled SO(3) rigid body resonant systems
We now derive a new resonant Euler system which couples two SO(3;R)
rigid bodies via a common principle axis of inertia and a common moment
of inertia. This 5-dimensional system conserves energy, helicity, and is rather
interesting in that dynamics on its homoclinic manifolds show bursting cas-
cades of enstrophy to the smallest scale in the resonant set. We consider the
equivalent periodic lattice geometry under the conditions of Proposition 4.1.
In Appendix, we prove that for an “irreducible” 3-wave resonant set which
now satisfies the algebraic “degeneracy” (A-4), there exist exactly two “prim-
itive” resonant triplets (k,m, n) and (k̃, m̃, n), where k, m, k̃, m̃ are relative
prime integer valued vectors in Z3:
Lemma 4.8 Under the algebraic degeneracy condition (A-4) the irreducible
equivariant family of projective lines in K∗ is exactly generated by the follow-
ing two “primitive” triplets:
n = k +m, k = ak, m = bm, (4.8a)
n = k̃ + m̃, k̃ = a′σi(k) + b
′σj(m), (4.8b)
that is,
n = ak + bm, (4.8c)
n = a′σi(k) + b
′σj(m), (4.8d)
where σi 6= σj are some reflection symmetries, a, b, a′, b′ are relatively
prime integers, positive or negative, and k, m are relatively prime integer
valued vectors in Z3, that is:
(a, a′) = (b, b′) = (a, b) = (a′, b′) = 1, (k,m) = 1,
where ( , ) denotes the Greatest Common Denominator of two integers. All
other resonant wave number triplets are generated by the group actions σl,
l = 1, 2, 3, and homothetic rescalings (k,m, n) → γ(k,m, n), (k̃, m̃, n) →
γ(k̃, m̃, n), (γ ∈ Z) of the “primitive” triplets.
Remark 4.9 It can be proven that the set of such coupled “primitive” triplets
is not empty on the periodic lattice. The algebraic irreducibility condition of
Lemma 4.2 implies that ±k3/|k| = ±k̃3/|k̃| and ±m3/|m| = ±m̃3/|m̃|, which
is obviously verified in equations (4.8).
Theorem 4.10 Under conditions of Lemma 4.8 the resonant Euler system
reduces to a system of two rigid bodies coupled via an(t):
ȧk = (λm − λn)Γaman (4.9a)
ȧm = (λn − λk)Γanak (4.9b)
ȧn = (λk − λm)Γakam + (λk̃ − λm̃)Γ̃ak̃am̃ (4.9c)
ȧm̃ = (λn − λk̃)Γ̃anak̃ (4.9d)
ȧk̃ = (λm̃ − λn)Γ̃am̃an, (4.9e)
where Γ = i < Φk×Φm,Φ∗n >, Γ̃ = i < Φk̃×Φm̃,Φ∗n >. Energy and Helicity
are conserved.
Theorem 4.11 The resonant system (4.9) possesses three independent con-
servation laws:
E1 = a2k + (1− α)a2m, (4.10a)
E2 = a2n + αa2m + (1 − α̃)a2m̃, (4.10b)
E3 = a2k̃ + α̃a
m̃, (4.10c)
where
α = (λm − λk)/(λn − λk), (4.11a)
α̃ = (λm̃ − λn)/(λk̃ − λn). (4.11b)
Theorem 4.12 Under the conditions
λm < λk < λn, (4.12a)
λm̃ < λn < λk̃, (4.12b)
which imply α < 0, α̃ < 0, the equilibria (±ak(0), 0, 0, 0,±ak̃(0)) are hyper-
bolic for |ak̃(0)| small enough with respect to |ak(0)|. The unstable manifolds
of these equilibria are one dimensional, and the nonlinear dynamics of system
(4.9) are constrained on the ellipse E1 (4.10a) for ak(t), am(t), the hyperbola
E3 (4.10c) for ak̃(t), am̃(t), and the hyperboloid E2 (4.10b) for am(t), am̃(t),
an(t).
Theorem 4.13 Let the 2-manifold E1 ∩E2 ∩E3 be coordinatized by (am, am̃).
On this 2-manifold, the resonant system (4.9) is Hamiltonian, and therefore
integrable. Its Hamiltonian vector field h is defined by
ιhω = Γ(λn − λk)
− Γ̃(λn − λk̃)
, (4.13)
where ιhω designates the inner product of the symplectic 2-form
dam ∧ dam̃
akanak̃
(4.14)
with the vector field h.
Proof of Theorem 4.13: Eliminating ak(t) via E1, an(t) via E2, ak̃(t) via
E3, the resonant system (4.9) reduces to:
ȧm = ±Γ(λn − λk)(E1 − (1− α)a2m)
2 (E2 − αa2m + (α̃− 1)a2m̃)
ȧm̃ = ±Γ̃(λn − λk̃)(E2 − αa
m + (α̃− 1)a2m̃)
2 (E3 − α̃a2m̃)
after changing the time variable into
(E1 − (1− α)a2m)
2 (E2 − αa2m + (α̃− 1)a2m̃)
2 (E3 − α̃a2m̃)
2 ds .
On each component of the manifold E1 ∩E2 ∩E3, the following functionals
are conserved:
H(am, am̃) = ± Γ̃(λn − λk̃)
(E1 − (1− α)a2m)1/2
± Γ(λn − λk)
(E3 − α̃a2m̃)1/2
Observe that the system of two coupled rigid bodies (4.9) does not seem to ad-
mit a simple Lie-Poisson bracket in the original variables (ak, am, an, am̃, ak̃).
Yet, when restricted to the 2-manifold E1 ∩E2 ∩E3 that is invariant under the
flow of (4.9), it is Hamiltonian and therefore integrable.
This raises the following interesting issue: according to the shadowing
Theorem 2.10, the Euler dynamics remains asymptotically close to that of
chains of coupled SO(3;R) and SO(3;C) rigid body systems. Perhaps some
new information could be obtained in this way. We are currently investigating
this question and will report on it in a forthcoming publication [G-M-N].
Already the simple 5-dimensional system (4.9) has interesting dynamical
properties, wich we could not find in the existing literature on systems related
to spinning tops.
Consider for instance the dynamics of the resonant system (4.9) with I.C.
topologically close to the hyperbola equilibria (±ak(0), 0, 0, 0,±ak̃(0)). Un-
der the conditions of (4.12) and with the help of the integrability Theorem
4.13, it is easy to construct equivariant families of homoclinic cycles at these
hyperbolic critical points:
Corollary 4.14 The hyperbolic critical points (±ak(0), 0, 0, 0,±ak̃(0)) pos-
sess 1-dimensional homoclinic cycles on the cones
a2n + (1− α̃)a2m̃ = −αa2m (4.15)
with α < 0, α̃ < 0.
Note that these are genuine homoclinic cycles, NOT sums of heteroclinic
connections. Initial conditions for the resonant system (4.9) are now chosen
in a small neighborhood of these hyperbolic critical points, the corresponding
orbits are topologically close to these cycles. With the ordering:
λm < λk < λn, (4.16a)
|λk| ≪ |λm|, |λk| ≪ λn, (4.16b)
λm̃ < λn < λk̃, (4.16c)
|λm̃| ≪ λk̃, (4.16d)
λk̃ ≫ λn, (4.16e)
which can be realized with |a
| ≫ 1 and | b
| ≪ 1 in the resonant triplets
(4.8), we can demonstrate bursting dynamics akin to Theorem 3.9 and 3.11 for
enstrophy and Hs norms, s ≥ 2. The interesting feature is the maximization
of |ak̃(t)| near the turning points of the homoclinic cycles on the cones (4.15).
This corresponds to transfer of energy to the smallest scale k̃, λk̃.
In a publication in preparation, we investigate infinite systems of the cou-
pled rigid bodies equations (4.9).
APPENDIX
We focus on a resonant wave number triplet (n, k,m) ∈ (Z∗)3 verifying
• the convolution relation
n = k +m, (A-1)
• the resonant 3-wave resonance relation
± n3√
1 + ϑ2n
2 + ϑ3n
± k3√
1 + ϑ2k
2 + ϑ3k
± m3√
1 + ϑ2m
2 + ϑ3m
(A-2)
• the condition of “non-catalyticity”
k3m3n3 6= 0, (A-3)
• and the degeneracy condition of [B-M-N4] (see p26)
Giri,j(k,m) = kinjml + klmjni = 0, (A-4)
where (i, j, l) is a permutation of (1, 2, 3).
Then, we know (see lemma 3.5 (2) of [B-M-N4]) that the system of equations
(A-3)-(A-4) for the unknown k and m, given the vector n, admits exactly 4
solutions in Z3 × Z3:
(k,m), (m, k), (k̃, m̃), (m̃, k̃).
Here k and m are the two vectors of the original resonant triplet, whereas
k̃ = ασi(k), m̃ = βσj(m)
where
mikl −mlki
mikl +mlki
/∈ {0,±1} and β = mlkj −mjkl
mlkj +mjkl
/∈ {0,±1}
and where the symmetries σi and σj are defined by
σi : u = (ul)l=1,2,3 →
(−1)δilul
l=1,2,3
One verifies that
σ2i = σ
j = Id, σiσj = σjσi = −σl.
That is, the group generated by σi and σj is the Klein group Z/2Z× Z/2Z.
Let us first write the irrational numbers α and β under the irreducible
representation
, β =
, with a, a′, b, b′ ∈ Z∗ and (a, a′) = (b, b′) = 1,
where ( , ) denotes the Greatest Common Denominator of the integer pair.
From k̃ ∈ Z3, it follows that a|a′k; but since (a, a′) = 1, the Euclid’s lemma
yields that a|k. Similarly, b|m. Now set
k ∈ Z3, m = 1
m ∈ Z3.
Hence the integer vector n admits the two decompositions
n = ak + bm = a′σi(k) + b
′σj(m).
Since the function
z 7−→ z3√
1 + ϑ2z
2 + ϑ3z
is homogeneous of degree 0, we see that within the resonance condition (A-2)
we can replace each vector k,m and n by any colinear vectors - either integer
or not. Suppose now that there exists some positive integer d 6= 1 such that
d|k; then d|n, so that by setting
n, k0 =
k, m0 =
we finally obtain
n0 = ak0 + bk0 = a
′σi(k0) + b
′σj(m0).
The triplets (n0, ak0, bm0) and (n0, a
′σi(k0), b
′σj(m0)) further verify from the
above remark, the convolution relation (A-1) and the resonance relation (A-2).
Hence, without loss of generality, we can assume that the only positive integer
d such that d|k and d|m is 1; which we denote by
(k,m) = 1.
Equivalently,
k1Z+ k2Z+ k3Z+m1Z+m2Z+m3Z+ = Z.
Finally, suppose there exists some positive integer d 6= 1 such that d|a and
d|b. Then d|n; set
n, a0 =
a, b0 =
Observe that
Giri,j(a0k, b0m) =
Giri,j(ak, bm) = 0.
It follows from lemma 3.5 (2) of [B-M-N4] that the vector n0 of the resonant
triplet (n0, a0k, b0m) can also be written as
n0 = k̂ + m̂ with (n0, k̂, m̂) verifying (A-2).
But then
n = dn0 = ak + bm = a
′σi(k) + b
′σj(m) = dk̂ + dm̂.
From lemma 3.5 (2) of [B-M-N4], (dk̂, dm̂) must coincide with either one of
the pairs
(a′σi(k), b
′σj(m)), (b
′σj(m), a
′σi(k)).
In particular, d|a′k and d|b′m. Since d|a and (a, a′) = 1, we have (d, a′);
similarly (d, b′) = 1. But then Euclid’s lemma yields that d|k and d|m, which
contradicts the fact that (k,m) = 1. Hence we have proven that (a, b) = 1.
In a similar way, one can show that (a′, b′) = 1.
Conclusion: It follows from the above study that n ∈ Z∗ admits the two
decompositions
n = ak + bm = a′σi(k) + b
′σj(m)
(a, a′) = (b, b′) = (a, b) = (a′, b′) = 1, (k,m) = 1.
The triplets (n, ak, bm) and (n, a′σi(k), b
′σj(m)) both verify the resonant
condition (A-2) (from the homogeneity of this condition) as well as the condi-
tion of non-catalyticity (A-3). Indeed, aba′b′ 6= 0 and the condition (A-3) on
the initial triplet (n, k,m) imply that the reduced triplet (n, k,m) also verifies
(A-3)). Finally, the degeneracy condition (A-4)
Giri,j(ak, bm) = 0
is verified.
Acknowledgments. We would like to thank A.I. Bobenko, C. Bardos
and G. Seregin for very useful discussions. The assistance of Dr. B. S. Kim
is gratefully acknowledged. A.M. and B.N. acknowledge the support of the
AFOSR contract FA9550-05-1-0047.
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	Introduction
	Vorticity waves and resonances of elementary swirling flows
	Strictly resonant Euler systems: the SO(3) case
	Strictly resonant Euler systems: the case of 3-waves resonances on small-scales
	 Infinite dimensional uncoupled SO(3) systems 
	Coupled SO(3) rigid body resonant systems
