Microlocal Asymptotic Analysis in Algebras of Generalized Functions
Microlocal Asymptotic Analysis in Algebras of Generalized
Functions
Antoine Delcroix
Equipe Analyse Algébrique Non Linéaire – Laboratoire A O C
Faculté des sciences - Université des Antilles et de la Guyane,
BP 250, 97157 Pointe à Pitre Cedex, Guadeloupe (France)
Antoine.Delcroix@univ-ag.fr
Jean-André Marti
Equipe Analyse Algébrique Non Linéaire – Laboratoire G T S I
Faculté des sciences – Université des Antilles et de la Guyane,
BP 250, 97157 Pointe à Pitre Cedex, Guadeloupe (France)
Jean-Andre.Marti@univ-ag.fr
Michael Oberguggenberger∗
Institute of Basic Sciences in Engineering – Unit of Engineering Mathematics
Faculty of Civil Engineering – University of Innsbruck,
Technikerstraße 13 – 6020 Innsbruck, Austria
Michael.Oberguggenberger@uibk.ac.at
November 6, 2018
Abstract
We introduce a new type of local and microlocal asymptotic analysis in algebras of gen-
eralized functions, based on the presheaf properties of those algebras and on the properties
of their elements with respect to a regularizing parameter. Contrary to the more classical
frequential analysis based on the Fourier transform, we can describe a singular asymptotic
spectrum which has good properties with respect to nonlinear operations. In this spirit we
give several examples of propagation of singularities through nonlinear operators.
Keywords: microlocal analysis, generalized functions, nonlinear operators, presheaf, propaga-
tion of singularities, singular spectrum.
Mathematics Subject Classification (2000): 35A18, 35A27, 46E10, 46F30, 46T30
1 Introduction
Various nonlinear theories of generalized functions have been developed over the past twenty
years, with contributions by many authors. These theories have in common that the space of
distributions is enlarged or embedded into algebras so that nonlinear operations on distribu-
tions become possible. These methods have been especially efficient in formulating and solving
nonlinear differential problems with irregular data.
Most of the algebras of generalized functions possess the structure of sheaves or presheaves,
which may contain some sub(pre)sheaves with particular properties. For example, the sheaf
∗Supported by FWF (Austria), grant Y237.
http://arxiv.org/abs/0704.1077v1
G of the special Colombeau algebras [2, 7, 15] contains the subsheaf G∞ of so-called regular
sections of G such that the embedding: G∞ → G is the natural extension of the classical one:
C∞ → D′. This notion of regularity leads to G∞-local or microlocal analysis of generalized
functions, extending the classical results on the C∞-microlocal analysis of distributions due
to Hörmander [8]. This concept has been slightly extended in [4] to less restrictive kinds
of measuring regularity. In [14], microlocal regularity theory in analytic and Gevrey classes
has been generalized to algebras of generalized functions. Many results on propagation of
singularities and pseudodifferential techniques have been obtained during the last years (see
[5, 6, 9, 10, 11]). Nevertheless, these results are still mainly limited to linear cases, since they
use frequential methods based on the Fourier transform.
In this paper, we develop a new type of asymptotic local and microlocal analysis of gener-
alized functions in the framework of (C, E ,P)-algebras [12, 13], following first steps undertaken
in [12]. An example of the construction is given by taking G as a special case of a (C, E ,P)-
structure (see Subsection 2.2 for details). Let F be a subsheaf of vector spaces (or algebras) of
G and (uε)ε a representative of u ∈ G (Ω) for some open set Ω ⊂ R
n. We first define OFG (u) as
the set of all x ∈ Ω such that uε tends to a section of F above some neighborhood of x. The
F-singular support of u is Ω\OFG (u). For fixed x and u, Nx(u) is the set of all r ∈ R+ such
that εruε tends to a section of F above some neighborhood of x. The F-singular spectrum of
u is the set of all (x, r) ∈ Ω × R+ such that r ∈ R+\Nx(u). It gives a spectral decomposition
of the F-singular support of u.
This asymptotic analysis is extended to (C, E ,P)-algebras. This gives the general asymptotic
framework, in which the net (εr)ε is replaced by any net a satisfying some technical conditions,
leading to the concept of the (a,F)-singular asymptotic spectrum. The main advantage is that
this asymptotic analysis is compatible with the algebraic structure of the (C, E ,P)-algebras.
Thus, the (a,F)-singular asymptotic spectrum inherits good properties with respect to nonlin-
ear operations (Theorem 15 and Corollary 16).
The paper is organized as follows. In Section 2, we introduce the sheaves of (C, E ,P)-algebras
and develop the local asymptotic analysis. Section 3 is devoted to the (a,F)-microlocal analysis
and specially to the nonlinear properties of the (a,F)-singular asymptotic spectrum. In Section
4 various examples of the propagation of singularities through non linear differential operators
are given.
2 Preliminary definitions and local parametric analysis
2.1 The presheaves of (C, E ,P)-algebras: the algebraic structure
We begin by recalling the notions from [12, 13] that form the basis for our study.
(a) Let:
(1) Λ be a set of indices;
(2) A be a solid subring of the ring KΛ (K = R or C); this means that whenever (|sλ|)λ ≤ (|rλ|)λ
for some ((sλ)λ, (rλ)λ) ∈ K
Λ ×A, that is, |sλ| ≤ |rλ| for all λ, it follows that (sλ)λ ∈ A ;
(3) IA be a solid ideal of A ;
(4) E be a sheaf of K-topological algebras over a topological space X .
Moreover, suppose that
(5) for any open set Ω in X, the algebra E(Ω) is endowed with a family P(Ω) = (pi)i∈I(Ω)
of semi-norms such that if Ω1, Ω2 are two open subsets of X with Ω1 ⊂ Ω2, it follows that
I(Ω1) ⊂ I(Ω2) and if ρ
1 is the restriction operator E(Ω2) → E(Ω1), then, for each pi ∈ P(Ω1)
the semi-norm p̃i = pi ◦ ρ
1 extends pi to P(Ω2) .
(6) Let Θ = (Ωh)h∈H be any family of open sets in X with Ω = ∪h∈HΩh. Then, for each
pi ∈ P(Ω), i ∈ I(Ω), there exist a finite subfamily of Θ: Ω1, . . . , Ωn(i) and corresponding
semi-norms p1 ∈ P(Ω1), . . . , pn(i) ∈ P(Ωn(i)), such that, for any u ∈ E(Ω)
pi (u) ≤ p1 (u |Ω1 ) + . . .+ pn(i)(u |Ωn(i)).
(b) Define |B| = {(|rλ|)λ , (rλ)λ ∈ B}, B = A or IA, and set
H(A,E,P)(Ω) =
(uλ)λ ∈ [E(Ω)]
| ∀i ∈ I(Ω), ((pi(uλ))λ ∈ |A|
J(IA,E,P)(Ω) =
(uλ)λ ∈ [E(Ω)]
| ∀i ∈ I(Ω), (pi(uλ))λ ∈ |IA|
C = A/IA,
Note that, from (2), |A| is a subset of A and that A+ = {(bλ)λ ∈ A, ∀λ ∈ Λ, bλ ≥ 0} = |A|.
The same holds for IA. Furthermore, (2) implies also that A is a K-algebra. Indeed, it suffices
to show that A is stable under multiplication by elements of K. Let c be in K and (aλ)λ ∈ A.
Then (caλ)λ satisfies (|caλ|)λ ≤ (|naλ|)λ for some n ∈ N. We have (naλ)λ ∈ A since A is stable
under addition. Thus, using (2), we get that (caλ)λ ∈ A.
For later reference, we recall the following notions entering in the definition of a sheaf A on
X. Let (Ωh)h∈H be a family of open sets in X with Ω = ∪h∈HΩh.
(F1) (Localization principle) Let u, v ∈ A(Ω). If all restrictions u|Ωh and u|Ωh , h ∈ H, coincide,
then u = v in A(Ω).
(F2) (Gluing principle) Let (uh)h∈H be a coherent family of elements of A(Ωh), that is, the
restrictions to the non-void intersections of the Ωh coincide. Then there is an element
u ∈ A(Ω) such that u|Ωh = uh for all h ∈ H.
Proposition 1 (i) H(A,E,P) is a sheaf of K-subalgebras of the sheaf E
(ii) J(IA,E,P) is a sheaf of ideals of H(A,E,P).
Proof. The proof can be found in [12, 13], so we just recall the main steps. We start from
the statement that E and EΛ are already sheaves of algebras. From (5), we infer that H(A,E,P)
and J(IA,E,P) are a presheaves (the restriction property holds) and that the localization property
(F1) is valid. To obtain the gluing property (F2) we need property (6), which generalizes the
situation from C∞ to E .
Theorem 2 The factor H(A,E,P)/J(IA,E,P) is a presheaf satisfying the localization principle
(F1).
Proof. From the previous proposition, we know that A = H(A,E,P)/J(IA,E,P) is a presheaf.
For Ω1 ⊂ Ω2, the restriction is defined by
A (Ω2)
−→ A (Ω1)
u 7−→ u |Ω1 := [uλ |Ω1 ]
where (uλ)λ is any representative of u ∈ A(Ω2) and [uλ |Ω1 ] denotes the class of (uλ |Ω1)λ.
The definition is consistent and independent of the representative because for each (uλ)λ∈Λ ∈
H(A,E,P)(Ω2) and (ηλ)λ∈Λ ∈ J(IA,E,P)(Ω2), we have
(uλ)λ |Ω1 := (uλ |Ω1 )λ ∈ H(A,E,P)(Ω1) , (ηλ)λ |Ω1 := (ηλ |Ω1 )λ ∈ J(IA,E,P)(Ω1)
The localization principle is also obviously fulfilled because J(IA,E,P) is itself a sheaf.
Proposition 3 Under the hypothesis (2), the constant sheaf H(A,K,|.|)/J(IA,K,|.|) is exactly the
ring C = A/IA.
Proof. We clearly have H(A,K,|.|) = A and J(IA,K,|.|) = IA.
Definition 1 The factor presheaf of algebras over the ring C = A/IA:
A = H(A,E,P)/J(IA,E,P)
is called a presheaf of (C, E ,P)-algebras.
Notation 1 We denote by [uλ] the class in A(Ω) defined by (uλ)λ∈Λ ∈ H(A,E,P)(Ω). For u ∈ A,
the notation (uλ)λ∈Λ ∈ u means that (uλ)λ∈Λ is a representative of u.
Remark 1 The problem of rendering A a sheaf (and even a fine sheaf) is not studied here. It
is well known that the Colombeau algebra G, which is a special case of a (C, E ,P)-algebra (see
Subsection 2.2), forms a fine sheaf [1, 7]. The sheaf property can be inferred from the existence
of a C∞-partition of unity associated to any open covering of an open set Ω of Rd. This existence
is fulfilled because X = Rd is a locally compact Hausdorff space. On the other hand, C∞ is a fine
sheaf because multiplication by a smooth function defines a sheaf homomorphism in a natural
way. Hence the usual topology and C∞-partition of unity defines the required sheaf partition of
unity. Observing that G is a sheaf of C∞-modules and using the well known result that a sheaf of
modules on a fine sheaf is itself a fine sheaf, we obtain the corresponding assertion about G. In
the general case, turning A into a sheaf requires additional hypotheses, which are not necessary
for the results in this paper. Indeed, the presheaf structure of A and the (F1)-principle are
sufficient to develop our local and microlocal asymptotic analysis.
Remark 2 The map ι : K → A defined by ι (r) = (r)λ is an embedding of algebras and induces
a ring morphism from K → C if, and only if, A is unitary (Lemma 14, [13]). Indeed, if A is
unitary, (r)λ = r (1λ)λ is an element of A since A is a K-algebra, and ι is clearly an injective
ring morphism. The converse is obvious. Moreover, if Λ is a directed set with partial order
relation ≺ and if
(7) IA ⊂
(aλ)λ ∈ A | lim
aλ = 0
then the morphism ι is injective. Indeed, if [ι (r)] = 0, relation (7) implies that the limit of the
constant sequence (r)λ is null, thus r = 0.
2.2 Relationship with distribution theory and Colombeau algebras
One main feature of this construction is that we can choose the triple (C, E ,P) such that the
sheaves C∞ and D′ are embedded in the corresponding sheaf A. In particular, we can multiply
(the images of) distributions in A.
We consider the sheaf E = C∞ over Rd, whereP is the usual family of topologies (PΩ)Ω∈O(Rd).
Here O
denotes the set of all open sets of Rd; this notation will be used in the sequel. Let
us recall that PΩ is defined by the family of semi-norms (pK,l)K⋐Ω,l∈N with
∀f ∈ C∞ (Ω) , pK,l (f) = sup
x∈K,|α|≤l
|∂αf (x)| .
From Lemma 14 in [13], it follows that the canonical maps, defined for any Ω ∈ O
σΩ : C
∞ (Ω) → H(A,E,P)(Ω) f 7→ (f)λ ,
are injective morphism of algebras if, and only if, A is unitary. Under this assumption, these
maps give rise to a canonical sheaf embedding of C∞ into H(A,E,P) and (using a partition of
unity in C∞ inducing a sheaf structure on A) to a canonical sheaf morphism of algebras from
C∞ into A. This sheaf morphism turns out to be a sheaf morphism of embeddings if Λ is a
directed set with respect to a partial order ≺ and if relation (7) holds.
We shall address the question of the embedding of D′ for the simple case of Λ = (0, 1]. For
a net (ϕε)ε of mollifiers given by
ϕε (x) =
, x ∈ Rd where ϕ ∈ D(Rd) and
ϕ (x) dx = 1,
and T ∈ D′
, the net (T ∗ ϕε)ε is a net of smooth functions in C
, moderately
increasing in
. This means that
(8) ∀K ⋐ Rd,∀l ∈ N, ∃m ∈ N : pK,l (T ∗ ϕε) = o(ε
−m), as ε→ 0.
This justifies to choose
(rε)ε ∈ R
(0,1] | ∃m ∈ N : |uε| = o(ε
−m), as ε→ 0
(rε)ε ∈ R
(0,1] | ∀q ∈ N : |uε| = o(ε
q), as ε→ 0
In this case (with E = C∞), the sheaf of algebras A = H(A,E,P)/J(IA,E,P) is exactly the so-called
special Colombeau algebra G [2, 7, 16]. Then, for all Ω ∈ O
, C∞ (Ω) is embedded in A (Ω)
σΩ : C
∞(Ω) → A(Ω) f 7→ [fε] with fε = f for all ε in (0, 1] ,
because the constant net (f)ε belongs to H(A,E,P)
and (f)ε ∈ J(IA,E,P) implies f = 0 in
C∞(Ω). Furthermore, D′
is embedded in A
by the mapping
ι : T 7→ (T ∗ ϕε)ε
Indeed, relation (8) implies that (T ∗ ϕε)ε belongs to H(A,E,P)
and (T ∗ ϕε)ε ∈ J(IA,E,P)
implies that T ∗ϕε → 0 in D
, as ε→ 0 and T = 0. Thus, ι is a well defined injective map.
With the help of cutoff functions, we can define analogously, for each open set Ω in Rd, an
embedding ιΩ of D
′ (Ω) into A (Ω), and finally a sheaf embedding D′ → A. This embedding
depends on the choice of the net of mollifiers (ϕε)ε. We refer the reader to [3, 15] for more
complete discussions about embeddings in Colombeau’s case and to [13] for the case of (C, E ,P)-
algebras.
2.3 An association process
We return to the general case with the assumption that A is unitary and Λ is a directed set
with partial order relation ≺ .
Let us denote by:
• Ω an open subset of X,
• F a given sheaf (or presheaf) of topological K-vector spaces (resp. K-algebras) over X
containing E as a subsheaf of topological algebras,
• a a map from R+ to A+ such that a(0) = 1 (for r ∈ R+, we denote a (r) by (aλ (r))λ).
In the Colombeau case, a typical example would be aε(r) = ε
r, ε ∈ (0, 1].
For (vλ)λ ∈ H(A,E,P) (Ω), we shall denote the limit of (vλ)λ for the F-topology by lim
F(Ω) vλ
when it exists. We recall that lim
F(V ) uλ |V = f ∈ F(V ) iff, for each F-neighborhood W of f ,
there exists λ0 ∈ Λ such that
λ ≺ λ0 =⇒ uλ|V ∈W.
We suppose also that we have, for each open subset V ⊂ Ω,
(9) J(IA,E,P)(V ) ⊂
(vλ)λ ∈ H(A,E,P)(V ) : lim
F(V ) vλ = 0
Definition 2 Consider u = [uλ] ∈ A(Ω), r ∈ R+, V an open subset of Ω and f ∈ F(V ). We
say that u is a (r)-associated with f in V :
F(V )
if lim
F(V ) (aλ (r) uλ |V ) = f.
In particular, if r = 0, u and f are called associated in V .
To ensure the independence of the definition with respect to the representative of u, we
must have, for any (ηλ)λ ∈ J(IA,E,P)(Ω), that lim
F(V ) aλ (r) ηλ |V = 0. As J(IA,E,P)(V ) is a
module over A, (aλ (r) ηλ |V )λ is in J(IA,E,P)(V ). Thus, our claim follows from hypothesis (9).
Example 1 Take X = Rd, F = D′, Λ =]0, 1], A = G, V = Ω, r = 0. The usual association
between u = [uε] ∈ G (Ω) and T ∈ D
′ (Ω) is defined by
u ∼ T ⇐⇒ u
D′(Ω)
T ⇐⇒ lim
D′(Ω) uε = T.
2.4 The F-singular support of a generalized function
We use the notations of Subsection 2.3. According to the hypothesis (9), we have, for any open
set Ω in X,
J(IA,E,P)(Ω) ⊂
(uλ)λ ∈ H(A,E,P)(Ω) : lim
F(V ) uλ = 0
FA(Ω) =
u ∈ A(Ω) | ∃ (uλ)λ ∈ u, ∃f ∈ F(Ω) : lim
F(V ) uλ = f
FA(Ω) is well defined because if (ηλ)λ belongs to J(IA,E,P)(Ω), we have lim
F(V ) ηλ = 0.
Moreover, FA is a sub-presheaf of vector spaces (resp. algebras) of A. Roughly speaking,
it is the presheaf whose sections above some open set Ω are the generalized functions of A (Ω)
associated with an element of F (Ω).
Thus, for u ∈ A (Ω), we can consider the set OFA (u) of all x ∈ Ω having an open neighbor-
hood V on which u is associated with f ∈ F (V ), that is:
OFA (u) = {x ∈ Ω | ∃V ∈ Vx : u |V ∈ FA(V )} ,
Vx being the set of all the open neighborhoods of x.
This leads to the following definition:
Definition 3 The F-singular support of u ∈ A(Ω) is denoted SFA (u) and defined as
SFA (u) = Ω\O
A (u) .
Remark 3 (i) The validity of the gluing principle (F2) is not necessary to get the notion of
support (and of F-singular support) of a section u ∈ A(Ω). More precisely, the localization
principle (F1) is sufficient to prove the following: The set
A (u) = {x ∈ Ω | ∃V ∈ Vx, u |V = 0}
is exactly the the union ΩA (u) of the open subsets of Ω on which u vanishes.
Indeed, (F1) allows to show that u vanishes on an open subset O of Ω if, and only if, it vanishes
on an open neighborhood of every point of O. This leads immediately to the required assertion.
Moreover, ΩA (u) = O
A (u) is the largest open set on which u vanishes, S
A (u) = Ω\O
A (u)
is exactly the support of u in its classical definition, and the F-singular support of u is a closed
subset of its support.
(ii) In contrast to the situation described above for the support, we need the gluing principle
(F2) if we want to prove that the restriction of u to O
A (u) belongs to FA(O
A (u)). We make
this precise in the following lemma.
Lemma 4 Take u ∈ A(Ω) and set ΩFA (u) = ∪i∈IΩi, (Ωi)i∈I denoting the collection of the open
subsets of Ω such that u |Ωi ∈ FA (Ωi). Then, if FA is a sheaf (even if A is only a prehesaf),
(i) ΩFA (u) is the largest open subset O of Ω such that u |O belongs to FA (O);
(ii) ΩFA (u) = O
A(u) and S
A (u) = Ω \ Ω
A (u).
Proof. (i) For i ∈ I, set u |Ωi = fi ∈ FA (Ωi). The family (fi)i∈I is coherent by assumption:
From (F2), there exists f ∈ FA(Ω
A (u)) such that f |Ωi = fi. But from (F1), we have f = u on
∪i∈IΩi = Ω
A (u). Thus u |ΩF
(u) ∈ FA(Ω
A (u)), and Ω
A (u) is clearly the largest open subset of
Ω having this property.
(ii) First, OFA (u) is clearly an open subset of Ω. For x ∈ O
A (u), set u |Vx = fx ∈ FA (Vx) for
some suitable neighborhood Vx. The open set O
A (u) can be covered by the family (Vx)x∈OF
As the family (fx) is coherent, we get from (F2) that there exists f ∈ FA
∪x∈OF
(u)Vx
such that
f |Vx = fx. From (F1), we have u = f on ∪x∈OF
(u)Vx and, therefore, u |OF
(u) ∈ FA(O
A (u)).
Thus OFA (u) is contained in Ω
A (u). Conversely, if x ∈ Ω
A (u), there exists an open neigh-
borhood Vx of x such that u |Vx ∈ FA (Vx). Thus x ∈ O
A (u) and the assertion (ii) holds.
Proposition 5 For any u, v ∈ A(Ω), if F is a presheaf of topological vector spaces, (resp.
algebras), we have:
SFA (u+ v) ⊂ S
A (u) ∪ S
A(v).
Moreover, in the resp. case, we have
SFA (uv) ⊂ S
A(u) ∪ S
A(v).
Proof. If x ∈ Ω belongs to OFA(u) ∩ O
A(v), there exist V and W in Vx such that u |V ∈
FA(V ) and v |W ∈ FA(W ). Thus (u + v)|V ∩W ∈ FA(V ∩W ) (resp. (uv)|V ∩W ∈ FA(V ∩W )),
which implies
OFA(u) ∩ O
A(v) ⊂ O
A(u+ v) (resp. O
A(u) ∩O
A(v) ⊂ O
A (uv) ).
The result follows by taking the complementary sets in Ω.
This proposition leads easily to the following:
Corollary 6 Let (uj)1≤j≤p be any finite family of elements in A(Ω). If F is a presheaf of
topological vector spaces, (resp. algebras), we have
SFA (
1≤j≤p
uj) ⊂
1≤j≤p
SFA(uj).
Moreover, in the resp. case, we have
SFA (
1≤j≤p
uj) ⊂
1≤j≤p
SFA(uj).
In particular, if uj = u for 1 ≤ j ≤ p, we have S
p) ⊂ SFA(u).
Example 2 Taking E = C∞; F = D′; A = G leads to the D′-singular support of an element of
the Colombeau algebra. This notion is complementary to the usual concept of local association
in the Colombeau sense. We refer the reader to [12, 13] for more details.
Example 3 In the following examples we consider X = Rd, E = C∞ and A = G.
(i) Take u ∈ σΩ (C
∞ (Ω)), where σΩ : C
∞ (Ω) → G (Ω) is the canonical embedding defined in
Subsection 2.2. Then SC
G (u) = ∅, for all p ∈ N.
(ii) Take ϕ ∈ D (R), with
ϕ (x) dx = 1, and set ϕε (x) = ε
−1ϕ (x/ε). As ϕε
D′(R)
δ, we have:
G ([ϕε]) = {0}. We note also that S
G ([ϕε]) = {0}. Indeed, for any K ⋐ R
∗ = R\ {0} and ε
small enough, ϕε is null on K and, therefore, ϕε
C∞(R∗)
(iii) Take u = [uε] with uε(x) = ε sin(x/ε). We have that lim pK,0(uε) = 0, for all K ⋐ R,
whereas lim pK,1(uε) does not exist for l ≥ 1. Therefore
G (u) = ∅ , S
G (u) = R.
Remark 4 For any (p, q) ∈ N
with p ≤ q, and u ∈ G, it holds that SC
G (u) ⊂ S
G (u).
3 The concept of (a,F)-microlocal analysis
Let Ω be an open set in X. Fix u = [uλ] ∈ A(Ω) and x ∈ Ω. The idea of the (a,F)-microlocal
analysis is the following: (uλ)λ may not tend to a section of F above a neighborhood of x,
that is, there exists no V ∈ Vx and no f ∈ F (V ) such that lim
F(V ) uλ = f . Nevertheless,
in this case, there may exist V ∈ Vx, r ≥ 0 and f ∈ F (V ) such that lim
F(V ) aλ(r)uλ = f ,
that is [aλ(r)uλ |V ] belongs to the subspace (resp. subalgebra) FA(V ) of A(V ) introduced in
Subsection 2.4. These preliminary remarks lead to the following concept.
3.1 The (a,F)-singular parametric spectrum
We recall that a is a map from R+ to A+ such that a(0) = 1 and F is a presheaf of topological
vector spaces (or topological algebras). For any open subset Ω of X, u = [uλ] ∈ A(Ω) and
x ∈ Ω, set
N(a,F),x (u) =
r ∈ R+ | ∃V ∈ Vx, ∃f ∈ F(V ) : lim
F(V ) (aλ(r)uλ |V ) = f
r ∈ R+ | ∃V ∈ Vx : [aλ (r)uλ |V ] ∈ FA(V )
It is easy to check that N(a,F),x (u) does not depend on the representative of u. If no confusion
may arise, we shall simply write
N(a,F),x (u) = Nx(u).
Theorem 7 Suppose that:
(a) For all λ ∈ Λ
∀ (r, s) ∈ R+, aλ(r + s) ≤ aλ(r)aλ(s),
and, for all r ∈ R+\ {0}, the net (aλ (r))λ converges to 0 in K.
(b) F is a presheaf of separated locally convex topological vector spaces.
Then we have, for u ∈ A(Ω):
(i) If r ∈ Nx(u), then [r,+∞) is included in Nx(u). Moreover, for all s > r, there exists V ∈ Vx
such that: lim
F(V ) (aλ(s)uλ |V ) = 0. Consequently, Nx(u) is either empty, or a sub-interval
of R+.
(ii) More precisely, suppose that for x ∈ Ω, there exist r ∈ R+, V ∈ Vx and f ∈ F(V ),
nonzero on each neighborhood of x included in V , such that lim
F(V ) (aλ(r)uλ |V ) = f . Then
Nx(u) = [r,+∞) .
(iii) In the situation of (i) and (ii), we have that 0 ∈ Nx(u) iff Nx(u) = R+. Moreover, if one
of these assertions holds, the limits lim
F(V ) (aλ (s) uλ |V ) can be non null only for s = 0.
Proof. (i) If r ∈ Nx(u), there exist V ∈ Vx and f ∈ F(V ) such that lim
F(V ) (aλ(r)uλ|V ) =
f. As F(V ) is locally convex, its topology may be described by a family QV = (qj)j∈J(V ) of
semi-norms. For all s > r, we have, for any j ∈ J (V ),
qj(aλ(s) (uλ |V )) = aλ(s) qj(uλ |V ) ≤ aλ(s− r) aλ(r) qj(uλ |V ) ≤ aλ(s − r) qj(aλ(r) uλ |V ).
From lim
qj (aλ(r) (uλ |V − f)) = 0, we have qj(aλ(r) uλ |V ) < +∞ and lim
qj (aλ(s)(uλ |V )) =
0, since aλ(s− r)
→ 0. Thus lim
F(V ) (aλ(s)uλ |V ) = 0.
(ii) From (i), we have [r,+∞) ⊂ Nx(u). Suppose that there exists t < r in Nx(u). Then we get
W ∈ Vx, which can be chosen included in V , and g ∈ F(W ) such that lim
F(W ) (aλ(t)uλ |W ) =
g. With the notations of the proof of (i), we have
qj(aλ(r) (uλ |W )) ≤ aλ(r − t)qj(aλ(t)uλ |W ).
As qj(aλ(t)uλ |V ) is bounded, it follows that lim
qj(aλ(r) (uλ |W )) = 0, which is in contradiction
with lim
F(V ) (aλ(r) (uλ |V ) = f 6≡ 0 on W.
(iii) The first assertion follows directly from (i) and the second from (ii).
From now on, we suppose that the hypotheses (a) and (b) of Theorem 7 are fulfilled. We
Σ(a,F),x(u) = Σx(u) = R+\Nx(u),
R(a,F),x (u) = Rx(u) = inf Nx(u).
According to the previous remarks and comments, Σ(a,F),x(u) is an interval of R+ of the form[
0, R(a,F),x (u)
0, R(a,F),x (u)
, the empty set, or R+.
Definition 4 The (a,F)-singular spectrum of u ∈ A(Ω) is the set
(a,F)
A (u) = {(x, r) ∈ Ω× R+ | r ∈ Σx(u)} .
Example 4 Take X = Rd, E = C∞, F = Cp (p ∈ N = N∪{+∞}), f ∈ C∞ (Ω). Set u =[(
and v =
ε−1 |ln ε| f
in A (Ω) = G (Ω). Then, for all x ∈ R,
N(a,Cp),x (u) = [1,+∞) , N(a,Cp),x (v) = (1,+∞) , R(a,Cp),x (u) = R(a,Cp),x (v) = 1.
Remark 5 We have: Σ(a,F),x(u) = ∅ iff N(a,F),x(u) = R+ and, according to Theorem 7, iff
0 ∈ N(a,F),x(u), that is, there exist (V, f) ∈ Vx×F(V ) such that lim
F(V ) (aλ(0)uλ |V ) = f . As
aλ(0) ≡ 1, this last assertion is equivalent to x ∈ O
A (u). Thus Σ(a,F),x(u) = ∅ iff x /∈ S
A (u).
This remark implies directly the:
Proposition 8 The projection of the (a,F)-singular spectrum of u on Ω is the F-singular
support of u.
3.2 Example: The Colombeau case
In this subsection we investigate the relationship between the (a,F)-singular spectrum and the
sharp topology for X = Rd, E = C∞, F = Cp (p ∈ N), A = G, aε (r) = ε
r. First, let us remark
that, for u = [uε] ∈ G (Ω), x ∈ Ω (Ω ∈ O
), N(a,Cp),x (u) is never empty.
Indeed, consider V ∈ Vx with V ⋐ Ω. There exists m > 0 such that pp,V (uε) = o (ε
−m) as
ε → 0. Thus, p
(uε) = o (ε
−m) for all k ≤ p and lim
Cp(V ) (ε
muε |V ) = 0. Thus [m,+∞) ⊂
N(a,Cp),x (u) .
Let us now recall the construction of the sharp topology on G (Ω) . For u = [(uε)ε] ∈ G (Ω),
K ⋐ Ω, l ∈ N, set
vK,l(u) = inf
r ∈ R
∣∣ pK,l (uε) = o(ε−r) as ε→ 0
The real number vK,l(u) is well defined, i.e. does not depend on the representative of u, and is
called the (K, l)-valuation of u. It has the usual properties:
(i) ∀λ ∈ C\{0}, ∀u ∈ G (Ω) , vK,l(λu) = vK,l(u) ;
(ii) ∀u, v ∈ G (Ω) , vK,l(u+ v) ≤ sup(vK,l(u), vK,l(v)).
The family (vK,l) permits to define the (K, l)-pseudodistances dK,l on G (Ω) by
∀ (u, v) ∈ G (Ω)
, dK,l (u, v) = exp (vK,l(u− v)) ,
which turns out to be ultrametric:
∀ (u, v, w) ∈ G (Ω)
, dK,l (u, v) ≤ sup(dK,l(u,w), dK,l(w, v)).
The topology defined by the family (dK,l)K,l is called the sharp topology on G (Ω).
As we are interested here in valuations greater or equal to 0, we set, for u ∈ G (Ω),
νK,l(u) = sup (vK,l(u), 0) .
We can define, for x ∈ Ω, the l-valuation of u at x by
νx,l(u) = inf
(u) |V ∈ V (x) , V relatively compact
and set, for any p ∈ N,
νpx(u) = sup
0≤l≤p
νx,l(u).
Proposition 9 For all p ∈ N, [uε] ∈ G (Ω) and x ∈ Ω, we have
νpx(u) = R(a,Cp),x (u) = infN(a,Cp),x (u) .
Proof. Take r > ν
x(u). Then, for any l with 0 ≤ l ≤ p, one has r > νx,l(u) and there exists
V ∈ V (x) , V relatively compact, such that v
(u) < r. Thus, p
(uε) = o(ε
−r), as ε → 0,
and lim
Cp(V ) (ε
ruε |V ) = 0, which implies that r > R(a,Cp),x (u) and ν
x(u) ≥ R(a,Cp),x (u).
Conversely, if r > R(a,Cp),x (u), there exists V ∈ V (x) such that lim
Cp(V ) (ε
ruε |V ) = 0. For
any relatively compact neighborhood W of x included in V , we get p
(uε) = o(ε
−r) and
r > v
(u) > νx,l(u). Thus, r ≥ ν
x(u) and ν
x(u) ≤ R(a,Cp),x (u).
3.3 Some properties of the (a,F)-singular parametric spectrum
Notation 2 For u = [uλ] ∈ A (Ω), lim
F(V ) (aλ(r)uλ |V ) ∈ F (V ) means that there exists
f ∈ F (V ) such that lim
F(V ) (aλ(r)uλ |V ) = f .
3.3.1 Linear properties
Proposition 10 For any u, v ∈ A(Ω), we have
(a,F)
A (u+ v) ⊂ S
(a,F)
A (u) ∪ S
(a,F)
A (v) .
Proof. Let r be in Nx(u) ∩Nx(v). Then there exist V ∈ Vx and W ∈ Vx such that
F(V ) (aλ(r)uλ |V ) ∈ F (V ) and lim
F(W ) (aλ(r) vλ |W ) ∈ F (W ) .
Thus lim
F(V ∩W ) (aλ(r) (uλ + vλ) |V ∩W ) ∈ F (V ∩W ) and r ∈ Nx(u+ v). Consequently,
Nx(u) ∩Nx(v) ⊂ Nx(u+ v).
We obtain the result by taking the complementary sets in R+.
Corollary 11 For any u, u0, u1 in A(Ω) with
(i) u = u0 + u1 (ii) S
(a,F)
A (u0) = ∅,
we have
(a,F)
A (u) = S
(a,F)
A (u1) .
Proof. Proposition 10 and condition (ii) give S
(a,F)
A (u) ⊂ S
(a,F)
A (u1). As (i) implies
u0 = u− u1, we obtain the converse inclusion, and thus the equality.
3.3.2 Differential properties
We suppose that F is a sheaf of topological differential vector spaces (resp. algebras), with
continuous differentiation, admitting E as a subsheaf of topological differential algebras. Then
the sheaf A is also a sheaf of differential algebras with, for any α ∈ Nd and u ∈ A (Ω),
∂αu = [∂αuλ] , where (uλ)λ is any representative of u.
The independence of ∂αu on the choice of representative follows directly from the definition of
J(IA,E,P).)
Proposition 12 Let u be in A(Ω). For all ∂α, α ∈ Nd, we have
(a,F)
αu) ⊂ S
(a,F)
A (u) .
Proof. Take u ∈ A(Ω), α ∈ Nd, x ∈ Ω, r ∈ Nx(u). There exists V ∈ Vx, f ∈ F (V ) such
F(V ) (aλ(r)uλ |V ) = f.
The continuity of ∂α implies that
F(V ) (aλ(r)∂
α uλ |V ) = ∂
Thus Nx(u) ⊂ Nx(∂
αu). The result is proved.
In the following two results we require that F is a sheaf of topological modules over E , in
addition. The proofs are straightforward.
Proposition 13 Let g be in E(Ω) and u in A(Ω). We have
(a,F)
A (gu) ⊂ S
(a,F)
A (u) .
Propositions 10, 12 and 13 finally imply:
Corollary 14 Let P (∂) =
|α|≤m
α be a differential polynomial with coefficients in E(Ω). For
any u ∈ A(Ω), we have
(a,F)
A (P (∂)u) ⊂ S
(a,F)
A (u) .
3.3.3 Nonlinear properties
Theorem 15 For given u and v ∈ A(Ω), let Di (i = 1, 2, 3) be the following disjoint sets:
D1 = S
A (u)�(S
A (u) ∩ S
A (v)) ; D2 = S
A (v)�(S
A (u) ∩ S
A (v)) ; D3 = S
A (u) ∩ S
A (v).
Then the (a,F)-singular asymptotic spectrum of uv verifies
(a,F)
A (uv) ⊂ {(x,Σx(u)), x ∈ D1} ∪ {(x,Σx(v)), x ∈ D2} ∪ {(x,Ex(u, v)), x ∈ D3}
where (for any x ∈ D3)
Ex(u, v) =
[0, supΣx(u) + supΣx(v)] if Σx(u) 6= R+ and Σx(v) 6= R+
R+ if Σx(u) = R+ or Σx(v) = R+
Proof. Suppose that x belongs to D1. Then x is not in S
A (v) and we have
Σx(v) = ∅, Nx(v) = R+.
If Nx(u) is not empty, let r be in Nx(u). As Nx(v) = R+, we have r ∈ Nx(v). Thus there exists
V ∈ Vx (resp. W ∈ Vx) such that [aλ(r)uλ |V ] ∈ FA(V ) (resp. [aλ(r)vλ |W ] ∈ FA(W )). As F
is a sheaf of topological algebras we have
[aλ(r) (uλvλ) |V ∩W ] ∈ FA(V ∩W ).
Thus, r belongs to Nx(uv). Therefore, we have proved that Σx(uv) ⊂ Σx(u). If Nx(u) is empty,
we have Σx(u) = R+ and the above inclusion is obviously fulfilled. For x in D2, the same proof
gives Σx(uv) ⊂ Σx(v).
Consider x in D3. Then, Σx(u) and Σx(v) are not empty. We suppose first that both of them
are not equal to R+. Set R = supΣx(u) and S = supΣx(v). If r > R, there exists r
′ ∈ Nx(u)
such that R < r′ < r and then, from the part (i) of Theorem 7, there exists V ∈ Vx such that
F(V ) (aλ(r)uλ |V ) = 0.
Similarly, if s > S, there exists W ∈ Vx such that
F(W ) (aλ(s) vλ |W ) = 0.
Then lim
F(V ∩W ) (aλ(r)aλ(s) (uλvλ) |V ∩W ) = 0. By expressing this limit in terms of semi-
norms, as in the proof of Theorem 7 and by using the inequality aλ(r+ s) ≤ aλ(r)aλ(s), we get
that lim
F(V ∩W ) (aλ(r + s) (uλvλ) |V ∩W ) = 0. Thus
[r + s,∞[ ⊂ Nx(uv) or [0, r + s[ ⊃ Σx(uv)
for any r > R and s > S. Thus
Σx(uv) ⊂ [0, R+ S] = [0, supΣx(u) + supΣx(v)].
If Σx(u) or Σx(v) is equal to R+, the obvious inclusion Σx(uv) ⊂ R+ gives the last result.
Corollary 16 For given u ∈ A(Ω) and p ∈ N∗, we have
(a,F)
(x,Hp,x(u)), x ∈ S
A (u)
where Hp,x(u) =
[0, p supΣx(u)] if Σx(u) 6= R+
R+ if Σx(u) = R+
Proof. When Σx(u) = R+, the result is obvious. Suppose now Σx(u) 6= R+. We shall prove
the result by induction. If p = 1, the result is a simple consequence of the definitions. Suppose
that the result holds for some p ≥ 1. Set v = up in the previous theorem. We have
D1 = S
A (u)�S
p) ; D2 = ∅ ; D3 = S
(a,F)
(x,Σx(u)), x ∈ S
A (u)�S
(x, [0, (p + 1) supΣx(u)]), x ∈ S
by using the induction hypothesis. It follows a fortiori that
(a,F)
(x, [0, (p + 1) supΣx(u)]), x ∈ S
A (u)
4 Applications to partial differential equations
In this section we shall compute various (a,F)-singular spectra of solutions to linear and nonlin-
ear partial differential equations. Throughout we shall suppose that Λ =]0, 1], X = Rd, E = C∞,
F = Cp (1 ≤ p ≤ ∞) or F = D′, aε(r) = ε
r. The results will hold for any (C, E ,P)-algebra
A = H(A,E,P)/J(IA,E,P)
such that (aε(r))ε ∈ A+ for all r ∈ R+ and property (9) holds.
Example 5 The (a,Cp)-singular spectrum of powers of the delta function. Given a mollifier
of the form
ϕε (x) =
, x ∈ Rd where ϕ ∈ D(Rd), ϕ ≥ 0 and
ϕ (x) dx = 1,
its class in A(Rd) defines the delta function δ(x) as an element of A(Rd). Its powers are given
by (m ∈ N)
Clearly, the C0-singular spectrum is given by
(a,C0)
0, [0,md]
Differentiating ϕm(x) and observing that for each derivative there is a point x at which it does
not vanish we see that
(a,Ck)
0, [0,md + k]
Example 6 The (a,D′)-singular spectrum of powers of the delta function. Given a test function
ψ ∈ D(Rd), we have ∫
ϕmε (x)ψ(x) dx =
εmd−d
ϕm(x)ψ(εx) dx,
(a,D′)
m) = ∅ for m = 1, S
(a,D′)
0, [0,md − d[
for m > 1.
4.1 The singular spectrum of solutions to linear hyperbolic equations
Consider the Cauchy problem for the d-dimensional linear wave equation
∂2t uε(x, t)−∆uε(x, t) = 0, x ∈ R
d, t ∈ R
uε(x, 0) = u0ε(x), ∂tuε(x, 0) = u1ε(x), x ∈ R
where u0ε, u1ε ∈ C
∞(Rd) represent elements u0, u1 of an algebra A(R
d) as outlined at the
beginning of this section. Under suitable assumptions on the ring A, the corresponding net of
classical smooth solutions represents a unique solution u in the algebra A(Rd+1); for example,
this holds in the Colombeau case [16]. Let t → E(t) ∈ C∞(R : E ′(Rd)) be the fundamental
solution of the Cauchy problem. Then
uε(·, t) =
E(t) ∗ εru0ε + E(t) ∗ ε
ru1ε.
If for some r ≥ 0 and u0 ∈ D
′(Rd),
εru0ε(x)ψ(x) dx → 〈u0, ψ〉
for all ψ ∈ D(Rd), then
εruε(x)ψ(x) dx = 〈E(t) ∗ ε
ru0ε, ψ〉 = 〈ε
ru0ε, Ě(t) ∗ ψ〉 → 〈u0, Ě(t) ∗ ψ〉
for all ψ ∈ D(Rd) and t ∈ R as well. We arrive at the following assertion.
Proposition 17 Assume that S
(a,D′)
A (u0) and S
(a,D′)
A (u1) are contained in R
d×I, where I = ∅,
I = [0, r[ or I = [0, r] for some r, 0 ≤ r ≤ ∞. Let u ∈ A(Rd+1) be the solution to the linear
wave equation (10). Then S
(a,D′)
A (u(·, t)) ⊂ R
d × I for all t ∈ R.
This upper bound may or may not be reached, depending on the effects of finite propagation
speed or the Huyghens principle in odd space dimension d ≥ 3. We just illustrate some of the
possible effects for the one-dimensional wave equation with powers of delta functions as initial
data. Thus we consider the problem
∂2t uε(x, t)− ∂
xuε(x, t) = 0, x ∈ R, t ∈ R
uε(x, 0) = c0ϕ
ε (x), ∂tuε(x, 0) = c1ϕ
ε (x), x ∈ R,
where ϕ is a mollifier as in Example 5 and c0, c1 ∈ R. The solution to (11) is given by
uε(x, t) =
ϕmε (x− t) + ϕ
ε (x+ t)
∫ x+t
ϕnε (y) dy.
We observe that uε(x, t) = 0 for sufficiently small ε when |x| > |t|, that is, outside the light
cone, and uε(x, t) = sign(t)
εn−1‖ϕn‖L1(R) for sufficiently small ε when |x| < |t|.
Example 7 If in equation (11) c0 6= 0, c1 = 0 then
(a,D′)
A (u) = {(x, t, r) : |x| = |t|, 0 ≤ r < m− 1}
with the provision that S
(a,D′)
A (u) = ∅ when m = 1. If in equation (11) c0 = 0, c1 6= 0 then
(a,D′)
A (u) = {(x, t, r) : |x| ≤ |t|, 0 ≤ r < n− 1}.
When both c0 and c1 are nonzero the singular spectrum is obtained as the union of the two
spectra. For the C0-singular spectrum the following results hold: If in equation (11) c0 6= 0,
c1 = 0 then
(a,C0)
A (u) = {(x, t, r) : |x| = |t|, 0 ≤ r ≤ m}.
If c0 = 0, c1 6= 0 then
(a,C0))
A (u) = {(x, t, r) : |x| < |t|, 0 ≤ r < n− 1} ∪ {(x, t, r) : |x| = |t|, 0 ≤ r ≤ n− 1}.
4.2 The singular spectrum of solutions to semilinear hyperbolic equations
In this subsection we study the paradigmatic case of a semilinear transport equation
∂tuε(x, t) + λ(x, t)∂xuε(x, t) = F (uε(x, t)), x ∈ R, t ∈ R
uε(x, 0) = u0ε(x), x ∈ R
where λ and F are smooth functions of their arguments. In this situation, the singular spectrum
of the initial data may be decreased or increased, depending on the function F . We observe
that by a change of coordinates we may assume without loss of generality that λ ≡ 0. In fact,
denote by s → γ(x, t, s) the characteristic curve of (12) passing through the point x at time
s = t, that is the solution to
γ(x, t, s) = λ(γ(x, t, s), s), γ(x, t, t) = x.
The function v(y, s) = u(γ(y, 0, s), s) is a solution of the initial value problem
∂sv(y, s) = F (v(y, s), v(y, 0) = u0(y),
at least as long as the characteristic curves exist.
Example 8 (The dissipative case) The equation
∂tuε(x, t) = −u
ε(x, t), x ∈ R, t > 0
uε(x, 0) = u0ε(x), x ∈ R
has the solution
uε(x, t) =
u0ε(x)√
2tu20ε(x) + 1
2t+ 1/u20ε(x)
When the initial data are given by a power of the delta function, u0ε(x) = ϕ
ε (x), the solution
formula shows that uε(x, t) is a bounded function (uniformly in ε) and supported on the line
{x = 0}. Thus uε(x, t) converges to zero in D
′(R×]0,∞[), and so
(a,D′)
A (u0) =
0, [0,m − 1[
(a,D′)
A (u) = ∅.
Example 9 The equation
∂tuε(x, t) =
1 + u2ε(x, t), x ∈ R, t > 0
uε(x, 0) = u0ε(x), x ∈ R
has the solution
uε(x, t) = u0ε(x) cosh t+
1 + u20ε(x) sinh t.
We first take a delta function as initial value, that is, u0ε(x) = ϕε(x). Then
uε(x, t)ψ(x, t) dxdt =
ϕ(x) cosh t+
ε2 + ϕ2(x) sinh t
ψ(εx, t) dxdt
ϕ(x) cosh t+ |ϕ(x)| sinh t
ψ(0, t) dxdt
for ψ ∈ D(R2). Thus in this case
(a,D′)
A (u0) = S
(a,D′)
A (u) = ∅.
On the other hand, taking the derivative of a delta function as initial value, u0ε(x) = ϕ
ε(x), a
similar calculation shows that
uε(x, t)ψ(x, t) dxdt =
ϕ(x) cosh t+
ε4 + (ϕ′)2(x) sinh t
ψ(εx, t) dxdt
and so
(a,D′)
A (u0) = ∅, S
(a,D′)
A (u) = {(0, t, r) : t > 0, 0 ≤ r < 1}.
The next example shows that it is quite possible for the singular spectrum to increase with
time.
Example 10 The equation
∂tuε(x, t) =
uε(x, t) + 1)
uε(x, t) + 1
, x ∈ R, t > 0
uε(x, 0) = u0ε(x), x ∈ R
has the solution
uε(x, t) =
u0ε(x) + 1
provided u0ε > −1 in which case the function on the right hand side of the differential equation
is smooth in the relevant region. To demonstrate the effect, we take a power of the delta function
as initial value, that is u0ε(x) = ϕ
ε (x). Then
(a,D′)
A (u0) = {(0, r) : 0 ≤ r < m− 1}, S
(a,D′)
A (u) = {(0, t, r) : t > 0, 0 ≤ r < me
t − 1}.
In situations where blow-up in finite time occurs, microlocal asymptotic methods allow to
extract information beyond the point of blow-up. This can be done by regularizing the initial
data and truncating the nonlinear term. We demonstrate this in a simple situation.
Example 11 Formally, we wish to treat the initial value problem
∂tu(x, t) = u
2(x, t), x ∈ R, t > 0
u(x, 0) = H(x), x ∈ R
where H denotes the Heaviside function. Clearly, the local solution u(x, t) = H(x)/(1− t) blows
up at time t = 1 when x > 0. Choose χε ∈ C
∞ (R) with
0 ≤ χε(z) ≤ 1 ; χε(z) = 1 if |z| ≤ ε
−s , χε(z) = 0 if |z| ≥ 1 + ε
−s , s > 0.
Further, let Hε(x) = H ∗ ϕε(x) where ϕε is a mollifier as in Example 5. We consider the
regularized problem
∂tuε(x, t) = χε
uε(x, t)
u2ε(x, t), x ∈ R, t > 0
uε(x, 0) = Hε(x), x ∈ R.
When x < 0 and ε is sufficiently small, uε(x, t) = 0 for all t ≥ 0. For x > 0, uε(x, t) = 1/(1− t)
as long as t ≤ 1− εs. The cut-off function is chosen in such a way that |χε(z)z
2| ≤ (1 + ε−s)2
for all z ∈ R. Therefore,
∂tuε ≤ (1 + ε
−s)2 always and ∂tuε = 0 when |uε| ≥ 1 + ε
Continuing the regularized solution beyond time t = 1 − εs, we infer by combining the two
inequalities that ε−s ≤ uε(x, t) ≤ 1 + ε
−s for t ≥ 1 − εs when x > 0 and ε is sufficiently
small. Finally, as long as t < 1, the regularized solution remains bounded with respect to ε near
(0, t) for ε small enough; after t = 1, the asymptotic growth of order ε−s spills over into any
neighborhood of every point (x, t) for x ≥ 0.
Collecting all previous estimates, we obtain the following C0-singular support and
singular spectrum (for aε(r) = ε
r) of u = [uε]:
A (u) = S1(u) ∪ S2(u) with S1(u) = {(0, t) : 0 ≤ t < 1} ; S2(u) = {(x, t) : x ≥ 0, t ≥ 1},
(a,C0)
A (u) = (S1(u)× {0}) ∪ (S2(u)× [0, s]) .
The C0-singularities (resp.
-singularities) of u are described by means of two sets: S1(u)
and S2(u) (resp. S1(u) × {0} and S2(u) × [0, s]). The set S1(u) (resp. S1(u) × {0}) is related
to the data C0 (resp.
)-singularity. The set S2(u) (resp. S2(u) × [0, s]) is related to the
singularity due to the nonlinearity of the equation giving the blow-up at t = 1. The blow-up locus
is the edge {x ≥ 0, t = 1} of S2(u) and the strength of the blow-up is measured by the length
s of the fiber [0, s] above each point of the blow-up locus. This length is closely related to the
diameter of the support of the regularizing function χε and depends essentially on the nature
of the blow-up: Changing simultaneously the scales of the regularization and of the cut-off (i.e.
replacing ε by some function h(ε) → 0 in the definition of ϕε and χε) does not change the fiber
and characterizes a sort of moderateness of the strength of the blow-up.
4.3 The strength of a singularity and the sum law
When studying the propagation and interaction of singularities in semilinear hyperbolic systems,
Rauch and Reed [18] defined the strength of a singularity of a piecewise smooth function. We
recall this notion in the one-dimensional case. Assume that the function f : R → R is smooth
on ] − ∞, x0] and on [x0,∞[ for some point x0 ∈ R. The strength of the singularity of f
at x0 is the order of the highest derivative which is still continuous across x0. For example,
if f is continuous with a jump in the first derivative at x0, the order is 0; if f has a jump
at x0, the order is −1. Travers [21] later generalized this notion to include delta functions.
Slightly deviating from her definition, but in line with the one of [18], we define the strength of
singularity of the k-th derivative of a delta function at x0, ∂
xδ(x− x0), by −k − 2.
The significance of these definitions is seen in the description of what Rauch and Reed
termed anomalous singularities in semilinear hyperbolic systems. We demonstrate the effect in
a paradigmatic example, also due to [18], the (3× 3)-system
(∂t + ∂x)u(x, t) = 0, u(x, 0) = u0(x)
(∂t − ∂x)v(x, t) = 0, v(x, 0) = v0(x)
∂tw(x, t) = u(x, t)v(x, t), w(x, 0) = 0
Assume that u0 has a singularity of strength n1 ≥ −1 at x1 = −1 and v0 has a singularity
of strength n2 ≥ −1 at x2 = +1. The characteristic curves emanating from x1 and x2 are
straight lines intersecting at the point x = 0, t = 1. Rauch and Reed showed that, in general,
the third component w will have a singularity of strength n3 = n1 + n2 + 2 along the half-ray
{(0, t) : t ≥ 1}. This half-ray does not connect backwards to a singularity in the initial data
for w, hence the term anomalous singularity. The formula n3 = n1 + n2 + 2 is called the sum
law. Travers extended this result to the case where u0 and v0 were given as derivatives of
delta functions at x1 and x2. We are going to further generalize this result to powers of delta
functions, after establishing the relation between the strength of a singularity of a function f
at x0 and the singular spectrum of f ∗ ϕε.
We consider a function f : R → R which is smooth on ]−∞, x0] and on [x0,∞[ for some point
x0 ∈ R; actually only the local behavior near x0 is relevant. We fix a mollifier ϕε(x) =
as in Example 5 and denote the corresponding embedding of D′(R) into the (C, E ,P)-algebra
A(R) by ι. In particular, ι(f) = [f ∗ ϕε].
If f is continuous at x0, then limε→0 f ∗ ϕε = f in C
0. If f has a jump x0, this limit does
not exist in C0, but limε→0 ε
rf ∗ ϕε = 0 in C
0 for every r > 0. We have the following result.
Proposition 18 Let x0 ∈ R. If f : R → R is a smooth function on ]−∞, x0] and on [x0,∞[
or f(x) = ∂kxδ(x − x0) for some k ∈ N, then the strength of the singularity of f at x0 is −n if
and only if
Σ(a,C1),x0
= [0, n].
Here n ∈ N and aε(r) = ε
Proof. When n = 0, the function f is continuous and its derivative has a jump at x0.
From what was said before Proposition 18 it follows that Σ(a,C1),x0
= {0}. When n = 1,
the function f has a jump itself at x0 and its distributional derivative contains a delta function
part. Thus limε→0 ε
rf ∗ϕε = 0 in C
0 for every r > 0 and limε→0 ε
r∂xf ∗ϕε = 0 in C
0 for every
r > 1, and neither of the two limits exists for smaller r. Therefore, Σ(a,C1),x0
= [0, 1].
When n ≥ 2, f(x) = ∂n−2x δ(x− x0) and the assertion is straightforward.
We shall now return to the model equation (13) and demonstrate that the sum law remains
valid when the initial data are powers of delta functions. We work in suitable (C, E ,P)-algebras
A(R) and A(R2) in which the initial value problem (13) can be uniquely solved (see the discus-
sion at the beginning of Subsection 4.1). We still consider the scale aε(r) = ε
Proposition 19 Let u0(x) = δ
m(x+1), v0(x) = δ
n(x−1) for some m,n ∈ N∗. Let w ∈ A(R2)
be the third component of the solution to problem (13). Then w(x, t) vanishes at all points (x, t)
with x 6= 0 as well as (0, t) with t < 1, and
Σ(a,C1),(0,t)
⊂ [0,m+ n]
for t ≥ 1.
Proof. A representative of w is given by
wε(x, t) =
ϕmε (x+ 1− s)ϕ
ε (x− 1 + s) ds.
The fact that the mollifier ϕ has compact support entails that wε(x, t) vanishes for sufficiently
small ε whenever x 6= 0 or t < 1. We have
wε(x, t) =
(x+ 1− s
(x− 1 + s
∂twε(x, t) =
(x+ 1− t
(x− 1 + t
∂xwε(x, t) =
εm+n+1
(x+ 1− s
(x+ 1− s
(x− 1 + s
εm+n+1
(x+ 1− s
(x− 1 + s
(x− 1− s
If the support of ϕ is contained in an interval [−κ, κ], say, then the t-integrations extend at
most from x+ 1− κε to x+ 1 + κε at fixed x. Therefore, all terms converge to zero uniformly
on R2 when multiplied by εr with r > m+ n. This proves the assertion.
Using the correspondence between the singular spectrum and the strength of a singularity
formulated in Proposition 18, as well as Example 5, we may say that the strength of the
singularity of δm(x + 1) at x0 = −1 is n1 = −m − 1, while the strength of the singularity of
δn(x−1) at x0 = +1 is n2 = −n−1. The strength of the singularity of the solution w at points
(0, t) with t ≥ 1 is −m− n = n1 + n2 + 2 and is seen to satisfy the sum law.
4.4 Regular Colombeau generalized functions
The subsheaf G∞ of regular Colombeau functions of the sheaf G is defined as follows [16]:
Given an open subset Ω of Rd, the algebra G∞(Ω) comprises those elements u of G(Ω) whose
representatives (uε)ε satisfy the condition
(14) ∀K ⋐ Ω ∃m ∈ N ∀l ∈ N : pK,l(uε) = o(ε
−m) as ε→ 0.
The decisive property is that the bound of order ε−m is uniform with respect to the order
of derivation on compact sets. The algebra G∞(Ω) satisfies G∞(Ω) ∩ D′(Ω) = C∞(Ω) and
forms the basis for the investigation of hypoellipticity of linear partial differential operators
in the Colombeau framework. We are going to characterize the G∞-property in terms of the
C∞-singular spectrum. The scale a is still given by aε(r) = ε
Proposition 20 Let u ∈ G(Ω). Then u belongs to G∞(Ω) if and only if
Σ(a,C∞),x
6= R+
for all x ∈ Ω.
Proof. If u ∈ G∞(Ω), x ∈ Ω and Vx is a relatively compact open neighborhood of x, property
(14) says that there is m ∈ N such that limε→0 ε
muε = 0 in C
∞(Vx). Thus Σ(a,C∞),x
6= R+.
Conversely, if Σ(a,C∞),x
6= R+ we can find an open neighborhood Vx of x and m(x) ∈ N
such that limε→0 ε
ruε = 0 in C
∞(Vx) for all r ≥ m. Any compact set K can be covered by
finitely many such neighborhoods. Letting m be the maximum of the numbers m(x) involved,
we obtain property (14).
In relation with regularity theory of solutions to nonlinear partial differential equations, a
further subalgebra of G(Ω) has been introduced in [17] – the algebra of Colombeau functions of
total slow scale type. It consists of those elements u of G(Ω) whose representatives (uε)ε satisfy
the condition
(15) ∀K ⋐ Ω ∀r > 0 ∀l ∈ N : pK,l(uε) = o(ε
−r) as ε→ 0.
The term slow scale refers to the fact that the growth is slower than any negative power of ε
as ε→ 0. This property can again be characterized by means of the singular spectrum.
Proposition 21 An element u ∈ G(Ω) is of total slow scale type if and only if
Σ(a,C∞),x
⊂ {0}
for all x ∈ Ω.
Proof. If u is of total slow scale type, x ∈ Ω and Vx is a relatively compact open neigh-
borhood of x, property (15) implies that limε→0 ε
suε = 0 in C
∞(Vx) for every s > 0. Thus
Σ(a,C∞),x
⊂ {0}. To prove the converse, we take a compact subset K and r > 0 and cover
K by finitely many neighborhoods Vx of points x ∈ K such that limε→0 ε
ruε = 0 in C
∞(Vx).
Then property (15) follows.
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	Introduction
	Preliminary definitions and local parametric analysis
	The presheaves of (C,E,P)-algebras: the algebraic structure 
	Relationship with distribution theory and Colombeau algebras
	An association process
	The F-singular support of a generalized function
	The concept of (a,F)-microlocal analysis
	The (a,F)-singular parametric spectrum
	Example: The Colombeau case
	Some properties of the (a,F)-singular parametric spectrum
	Linear properties
	Differential properties
	Nonlinear properties
	Applications to partial differential equations
	The singular spectrum of solutions to linear hyperbolic equations
	The singular spectrum of solutions to semilinear hyperbolic equations
	The strength of a singularity and the sum law
	Regular Colombeau generalized functions
