Annales de la Faculté des Sciences de Toulouse

Mario Wschebor

Smoothing and occupation measures of stochastic processes

Murray Marshall

Representations of non-negative polynomials having finitely many zeros

Hamidou Dathe

Jean-François Quint

Exemples de feuilletages de Lie

François Berteloot

Méthodes de changement d’échelles en analyse complexe

Laurent Danielo

Construction de métriques d’Einstein à partir de transformations biconformes

Raphaël Danchin

The inviscid limit for density-dependent incompressible fluids

Ian M. Aberbach

Florian Enescu

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
IAN M. ABERBACH & FLORIAN ENESCU
When does the F -signature exist?
Tome XV, no 2 (2006), p. 195-201.
<http://afst.cedram.org/item?id=AFST_2006_6_15_2_195_0>
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Annales de la Faculté des Sciences de Toulouse Vol. XV, n◦ 2, 2006
pp. 195–201
When does the F-signature exists?(∗)
Ian M. Aberbach (1), Florian Enescu (2)
ABSTRACT. — We show that the F -signature of an F -finite local ring R of
characteristic p > 0 exists when R is either the localization of an N-graded
ring at its irrelevant ideal or Q-Gorenstein on its punctured spectrum.
This extends results by Huneke, Leuschke, Yao and Singh and proves the
existence of the F -signature in the cases where weak F -regularity is known
to be equivalent to strong F -regularity.
RÉSUMÉ. — Nous prouvons dans cet article l’existence de la F -signature
d’un anneau local F -fini R, de caractéristique positive p, quand R est la
localisation à l’unique idéal homogène maximal d’un anneau N-gradué
ou quand R est Q-Gorenstein sur son spectre épointé. Ceci généralise les
résultats de Huneke, Leuschke, Yao et Singh et prouve l’existence de la
F -signature dans les cas où faible et forte F -régularité sont équivalentes.
1. A sufficient condition for the existence of the F-signature
Let (R,m, k) be a reduced, local F -finite ring of positive characteristic
p > 0 and Krull dimension d. Let
R1/q = Raq ⊕Mq
be a direct sum decomposition of R1/q such that Mq has no free direct sum-
mands. If R is complete, such a decomposition is unique up to isomorphism.
Recent research has focused on the asymptotic growth rate of the numbers
aq as q → ∞. In particular, the F -signature (defined below) is studied in [7]
and [3], and more generally the Frobenius splitting ratio is studied in [2].
(∗) Reçu le 7 juin 2004, accepté le 15 décembre 2004
(1) Department of Mathematics, University of Missouri, Columbia, MO 65211.
E-mail: aberbach@math.missouri.edu
(2) Department of Mathematics and Statistics, Georgia State University, Atlanta,
30303 and The Institute of Mathematics of the Romanian Academy (Romania).
E-mail: fenescu@mathstat.gsu.edu
Aberbach was partially supported by a grant from the NSA.
– 195 –
Ian M. Aberbach, Florian Enescu
For a local ring (R,m, k) , we set α(R) = logp[kR : k
p
R]. It is easy to
see that, for an m-primary ideal I of R, λ(R1/q/IR1/q) = λ(R/I [q])/qα(R),
where λ(−) represents the length function over R.
We would like to first define the notion of F -signature as it appears in [3]
and [7].
Definition 1.1. — The F -signature of R is s(R) = limq→∞
aq
qd+α(R)
, if
it exists.
The following result, due to Aberbach and Leuschke [3], holds:
Theorem 1.2. — Let (R,m, k) be a reduced Noetherian ring of positive
characteristic p. Then lim infq→∞ aq/qd+α(R) > 0 if and only if
lim supq→∞ aq/qd+α(R) > 0 if and only if R is strongly F -regular.
The question of whether or not, in a strongly F -regular ring, s(R) exists,
is open. We show in this paper that its existence is closely connected to the
question of whether or not weak and strong F -regularity are equivalent.
Smith and Van den Bergh ([10]) have shown that the F -signature of R
exists when R has finite Frobenius representation type (FFRT) type, that
is, if only finitely many isomorphism classes of indecomposable maximal
Cohen-Macaulay modules occur as direct summands of R1/q for any q = pe.
Yao has proven that, under mild conditions, tight closure commutes with
localization in a ring of FFRT type, [11]. Moreover, Huneke and Leuschke
proved that if R is also Gorenstein, then the F -signature exists, [7]. Yao
has recently extended this result to rings that are Gorenstein on their punc-
tured spectrum, [12]. Singh has also shown that the F -signature exists for
monomial rings, [9].
Let (R,m) be an approximately Gorenstein ring. This means that R has
a sequence of m-primary irreducible ideals {It}t cofinal with the powers of m.
By taking a subsequence, we may assume that It ⊃ It+1. For each t, let ut be
an element of R which represents a socle element modulo It. Then there is,
for each t, a homomorphism R/It ↪→ R/It+1 such that ut+It → ut+1+It+1.
The direct limit of the system will be the injective hull E = ER(R/m) and
each ut will map to the socle element of E, which we will denote by u.
Hochster has shown that every excellent, reduced local ring is approximately
Gorenstein ([5]).
– 196 –
When does the F-signature exists?
Aberbach and Leuschke have shown that, for every q, there exists t0(q),
such that
aq/(qd+α(R)) = λ(R/(I
[q]
t : u
q
t ))/q
d,
for all t  t0(q) (see [3], p. 55).
The situation when t0(q) can be chosen independently of q is of special
interest.
Definition 1.3. — We say that R satisfies Condition (A), if there exist
a sequence of irreducible m-primary ideals {It} and a t0 such that, for all
t  t0 and all q
(I [q]t : u
q
t ) = (I
[q]
t0 : u
q
t0).
Proposition 1.4. — Let (R,m, k) be a local reduced F -finite ring. If R
satisfies Condition A, then the F -signature exists.
Proof. — We know that R is approximately Gorenstein and hence we
will use the notation fixed in the paragraph above.
As explained above, Condition A implies that there exists t0, indepen-
dent of q, such that
aq/(qd+α(R)) = λ(R/(I
[q]
t0 : u
q
t0))/q
d,
for all q.
But λ(R/(I [q]t0 : u
q
t0)) = λ(R/I
[q]
t0 ) − λ(R/(It0 + ut0R)[q]). Dividing by
qd and taking the limit as q → ∞ yields s(R) = eHK(It0 , R) − eHK(It0 +
ut0R,R). 
Now we would like to concentrate on another condition, Condition (B),
that appeared first in the work of Yao. First we need to introduce some
notation.
Assume that E is the injective hull of the residue field k. By R(e) we
denote the R-bialgebra whose underlying abelian group equals R and the left
and right R-multiplication is given by a · r ∗ b = arbq, for a, b ∈ R, r ∈ R(e).
Let k = Ru → E be the natural inclusion and consider the natural
induced map φe : R(e) ⊗R E → R(e) ⊗R (E/k). Then aq/qα(R) = λ(ker(φe))
(by Aberbach-Enescu, Corollary 2.8 in [2], see also Yao’s work [12]).
– 197 –
Ian M. Aberbach, Florian Enescu
One can in fact see that
λ(ker(φe)) = λ(R/(c ∈ R : c⊗ u = 0 in R(e) ⊗R E)) = λ(R/ ∪t (I [q]t : uqt )).
Definition 1.5. — We say that R satisfies Condition (B) if there exists
a finite length submodule E′ ⊂ E such that, if
ψe : R(e) ⊗R E′ → R(e) ⊗R E′/k, then λ(ker(φe)) = λ(ker(ψe)), for all e.
Yao [12] has shown that Condition (B) implies that the F -signature of
R exists.
Proposition 1.6. — Let (R,m, k) be a local reduced F -finite ring. Then
Conditions (A) and (B) are equivalent.
Proof. — Assume that Condition (A) holds. Then one can take
E′ = R/It0 and then Condition (B) follows.
If Condition (B) holds, then take t0 large enough such that
E′ ⊂ Im(R/It0 → E).
As noted above, one can compute the length of the kernel of ψe as the
colength of {c ∈ R : c⊗u = 0 in R(e) ⊗RE′}. Since R/It0 injects into E we
see that {c ∈ R : c ⊗ u = 0 in R(e) ⊗R E′} is a subset of {c ∈ R : c ⊗ u =
0 in R(e) ⊗R R/It0} = (I
[q]
t0 : u
q
t0).
Since (I [q]t0 : u
q
t0) ⊂ (I
[q]
t : u
q
t ) for all t  t0, we see that Condition (B)
implies that (I [q]t0 : u
q
t0) = (I
[q]
t : u
q
t ) for all t  t0, which is Condition (A).

2. N-Graded rings
Let (R,m) be a Noetherian N-graded ring R = ⊕n0Rn, where R0 = k
is an F -finite field of characteristic p > 0.
For any graded R-module M one can define a natural grading on
R(e)⊗M : the degree of any tensor monomial r⊗m equals deg(r)+q deg(m).
In what follows we will need the following important Lemma by Lyubeznik
and Smith ([8], Theorem 3.2):
– 198 –
When does the F-signature exists?
Lemma 2.1. — Let R be an N-graded ring and M,N two graded R-
modules. Then there exists an integer t depending only on R such that
whenever
M → N
is a degree preserving map which is bijective in degrees greater than s, then
the induced map
R(e) ⊗M → R(e) ⊗N
is bijective in degrees greater than pe(s+ t).
Let E be the injective hull of Rm. In fact, E is also the injective hull of
R/m over R and as a result is naturally graded with socle in degree 0. We
can write E = ⊕n0En.
Let t be as in Lemma 2.1, and let s  −t − 1. Obviously the map
E′ = ⊕sn0En → E = ⊕n0En is bijective in degrees greater than s. So
by Lemma 2.1, the map R(e) ⊗E′ → R(e) ⊗E is bijective in degrees greater
than pe(s+ t)  −pe.
Theorem 2.2. — Let R be an N-graded reduced ring over an F -finite
field k of positive characteristic. Then Condition (B) is satisfied by R and
hence the F -signature of R exists.
Proof. — Let E be the injective hull of k = R/m over Rm. As above,
E = ⊕n0En, where 0 is the degree of the socle generator u of E.
Using the notation introduced above, we will let s = −t − 1 and
E′ = ⊕snn0En → E. So, R(e) ⊗ E′ → R(e) ⊗ E is bijective in degrees
greater than −pe. In particular it is bijective in degrees greater or equal
to 0.
We have the following exact sequences:
0 → k = Ru → E → E/k → 0
and
0 → k = Ru → E′ → E′/k → 0.
After tensoring with R(e), we get the exact sequences
R(e) ⊗ k = R(e) ⊗Ru → R(e) ⊗ E φe→ R(e) ⊗ E/k → 0
and
R(e) ⊗ k = R(e) ⊗Ru → R(e) ⊗ E′ ψe→ R(e) ⊗ E′/k → 0.
– 199 –
Ian M. Aberbach, Florian Enescu
One can easily see that ker(φe) and ker(ψe) are the submodules gener-
ated by 1 ⊗ u in R(e) ⊗ E and R(e) ⊗ E′, respectively.
The degree of 1⊗ u is q · 0 = 0 and we have noted that the natural map
R(e) ⊗ E′ → R(e) ⊗ E is bijective in degrees greater than −pe. This shows
that ker(φe)  ker(ψe) and hence Condition (B) is satisfied. 
3. Q-Gorenstein rings
We turn now to showing that Condition (A) holds in strongly F -regular
local rings which are Q-Gorenstein on the punctured spectrum. Let (R,m, k)
be such a ring of dimension d, and assume that R has a canonical module
(e.g. R is complete). In this case R has an unmixed ideal of height 1, say
J ⊆ R, which is a canonical ideal. We may pick an element a ∈ J which
generates J at all minimal primes of J , and then an element x2 ∈ m which
is a parameter on R/J such that x2J ⊆ aR. It is easy to see that then
xnJ (n) ⊆ anR for all n  1 (where J (n) is the height one component of Jn).
The condition that R is Q-Gorenstein on the punctured spectrum implies
that there is an integer h and two sequences of elements x3, . . . , xd ∈ m and
a3, . . . , ad ∈ J (h) such that xiJ (h) ⊆ aiR for 3  i  d, and x2, . . . , xd is a
s.o.p. on R/J . We may then pick x1 ∈ J such that x1, . . . , xd is an s.o.p. for
R. See [1], section 2.2 for more detail. Then by [1], Lemma 2.2.3 we have
that for any N  0 and any n  0,
(J (nh), xN2 , . . . , x̂Ni , . . . , x
N
d ) : x
∞
i = (J
(nh), xN2 , . . . , x̂
N
i , . . . , x
N
d ) : x
n
i .
(3.1)
Theorem 3.1. — Let (R,m, k) be an F -finite strongly F -regular ring
which is Q-Gorenstein on the punctured spectrum. Then R satisfies Condi-
tion (A). In particular the F -signature of R exists.
Proof. — If R is not complete, we observe that, since R is excellent, R̂
is strongly F -regular and Q-Gorenstein on the punctured spectrum. If {It}
is a sequence of ideal in R̂ showing condition (A) in R̂, then {It ∩R} does
so for R. Thus we will assume that R is complete.
Let J , h, and x1, . . . , xd be as discussed above. Let It = (xt−11 J, x
t
2, . . . , x
t
d).
Since xn1J ∼= J as R-modules, the quotient R/xn1J is Gorenstein. The
hypothesis that x2, . . . , xd are parameters on R/J and R/x1R (hence on
R/xn1J) then shows that It is irreducible (see [4], Proposition 3.3.18). The
sequence {It} is then a sequence of m-primary irreducible ideals cofinal with
– 200 –
When does the F-signature exists?
the powers of m. If u1 represents the socle element of I1, then we may take
ut = (x1 · · ·xd)t−1u1 to represent the socle element of It. We will show that
t0 may be taken to be 3.
Suppose that c ∈ I [q]t : uqt for some q. We will show that c ∈ I
[q]
3 : u
q
3.
Raising to the q′th power we have cq
′
uqq
′
t = cq
′ (
(x1 · · ·xd)t−1u1
)qq′ ∈ I [qq′]t
= (xt−11 J, x
t
2, . . . , x
t
d)
[qq′]. Hence cq
′ (
(x2 · · ·xd)t−1u1
)qq′ ∈ (xt2, . . . , xtd)[qq′] :
x
(t−1)qq′
1 + (J, x
t
2, . . . , x
t
d)
[qq′] = (J, xt2, . . . , x
t
d)
[qq′].
Write qq′ = nq′h+rq′ with 0  rq′ < h. Repeated application of equation
3.1 (using 1 rather than h for x2) gives
cq
′
((x2 · · ·xd)u1)qq
′ ∈ (J (nq′h), x2qq
′
2 , . . . , x
2qq′
d ). (3.2)
Let d ∈ J (h) ⊆ J (rq′ ). Multiplying by xqq
′
2 and using that x
qq′
2 J
(qq′) ⊆
aqq
′
R ⊆ J [qq′] we have dcq′
(
(x2 · · ·xd)2u1
)qq′ ∈ (J, x32, . . . , x3d)[qq′]. Multi-
plying by x2qq
′
1 shows that dc
q′uqq
′
3 = d(cu
q
3)
q′ ∈ (I [q]3 )[q
′]. Thus cuq3 ∈
(I [q]3 )
∗ = I [q]3 , as desired. 
Bibliography
[1] Aberbach (I. M.). — Some conditions for the equivalence of weak and strong
F -regularity, Comm. Alg. 30, 1635-1651 (2002).
[2] Aberbach (I. M.), Enescu (F.). — The structure of F -pure rings, Math. Zeit.,
to appear.
[3] Aberbach (I. M.), Leuschke (G.). — The F -signature and strong F -regularity,
Math. Res. Lett. 10, 51-56 (2003).
[4] Bruns (W.), Herzog (J.). — Cohen-Macaulay Rings, Cambridge University
Press, Cambridge (1993).
[5] Hochster (M.). — Cyclic purity versus purity in excellent Noetherian rings,
Trans. Amer. Math. Soc. 231, no. 2, 463-488 (1977).
[6] Hochster (M.) and Huneke (C.). — Tight closure and strong F -regularity,
Meḿoire no. 38, Soc. Math. France, 119-133 (1989).
[7] Huneke (C.), Leuschke (G.). — Two theorems about maximal Cohen-Macaulay
modules, Math. Ann. 324, 391-404 (2002).
[8] Lyubeznik (G.), Smith (K.E.). — Strong and weak F -regularity are equivalent
for graded rings, Amer. J. Math. 121, 1279-1290 (1999).
[9] Singh (A.K.). — The F -signature of an affine semigroup ring, J. Pure Appl. Al-
gebra, 196, 313-321 (2005).
[10] Smith (K.E.), Van den Bergh (M.). — Simplicity of rings of differential opera-
tors in prime characteristic, Proc. London. Math. Soc. (3) 75, no. 1, 32-62 (1997).
[11] Yao (Y.). — Modules with finite F -representation type, Jour. London Math. Soc.,
to appear.
[12] —, Observations on the F -signature of local rings of characteristic p > 0, preprint
2003.
– 201 –


When does the $F$-signature exist?

Mohamed Maliki

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
MOHAMED MALIKI
Continuous dependence of the entropy solution of general parabolic
equation
Tome XV, no 3 (2006), p. 589-598.
<http://afst.cedram.org/item?id=AFST_2006_6_15_3_589_0>
© Annales de la faculté des sciences de Toulouse Mathématiques,
2006, tous droits réservés.
L’accès aux articles de la revue « Annales de la faculté des sci-
ences de Toulouse, Mathématiques » (http://afst.cedram.org/), implique
l’accord avec les conditions générales d’utilisation (http://afst.cedram.
org/legal/). Toute reproduction en tout ou partie cet article sous quelque
forme que ce soit pour tout usage autre que l’utilisation à fin strictement
personnelle du copiste est constitutive d’une infraction pénale. Toute copie
ou impression de ce fichier doit contenir la présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Annales de la Faculté des Sciences de Toulouse Vol. XV, n◦ 3, 2006
pp. 589–598
Continuous dependence of the entropy solution
of general parabolic equation(∗)
Mohamed Maliki(1)
ABSTRACT. — We consider the general parabolic equation :
ut−∆b(u)+div F (u) = f in Q =]0, T [×IRN , T > 0 with u0 ∈ L∞(IRN ),
for a.e t ∈]0, T [, f(t) ∈ L∞(IRN ) and
∫ T
0
‖f(t)‖
L∞(IRN )
dt <∞.
We prove the continuous dependence of the entropy solution with respect
to F, b, f and the initial data u0 of the associated Cauchy problem.
This type of solution was introduced and studied in [MT3]. We start by
recalling the definition of weak solution and entropy solution. By applying
an abstract result (Theorem 2.3), we get the continuous dependance of
the entropy solution. The contribution of the present work consists of
considering the equation in the whole space IRn instead of a bounded
domain and considering a bounded data instead of integrable data.
RÉSUMÉ. — On considère l’équation parabolique générale :
ut − ∆b(u) + div F (u) = f dans Q =]0, T [×IRN , T > 0 avec u0 ∈
L∞(IRN ), f ∈ L1Loc(Q) pour p.p t ∈]0, T [ f(t) ∈ L∞(IRN ),
et
∫ T
0
‖f(t)‖
L∞(IRN )
dt <∞.
On montre la dépendance continue de la solution entropique du problème
de Cauchy associé, par rapport aux données F , b f et la donnée initiale
u0. Ce type de solution a été introduit et étudié dans [MT3].
On commence le travail par un rappel de la définition de la solution faible
et entropique ainsi que les résultats importants obtenus dans [MT3]. En-
suite on montre le résultat principal du travail en utilisant le lemme ab-
strait (Théorème 2.3). La contribution du travail est de traiter le problème
dans IRN ainsi que de considérer des données bornées au lieu des données
intégrables utilisées dans la littérature.
(∗) Reçu le 7 décembre 2004, accepté le 20 janvier 2006
(1) Université Hassan II, F.S.T. Mohammedia, B.P. 146 Mohammedia, Maroc.
E-mail: mohamedmaliki@yahoo.fr
– 589 –
Mohamed Maliki
1. Introduction and notations
We consider the Cauchy Problem (CP ) = (CP )(b, F, f, u0) :


ut − ∆b(u) + div F (u) = f in Q
u(0, .) = u0 in IRN ,
(CP )
where F ∈ C(IR, IRN ) , b ∈ C(IR), b such that F (0) = 0 and b(0) = 0.
We denote by Q =]0, T [×IRN with T > 0, and:


1) u0 ∈ L∞(IRN ),
2) f ∈ L1Loc(Q) for a.e t ∈]0, T [ f(t) ∈ L∞(IRN ),
3)
∫ T
0
‖f(t)‖
L∞(IRN )dt < ∞.
(H1)
(CP )(b, F, f, u0) is a degenerate second order problem. It is known there
is no classical solution and the weak solution is not unique. A vast literature
exists on the degenerate parabolic equation we consider. In this literature,
many results are proved about existence of weak solution, and uniqueness
under various additional conditions [AL], [BG], [BT1], [BW], [BT2], [DT]
and [YJ].
In our first work [MT3] while studying the existence and the uniqueness
of the entropy solution, we introduced sufficient conditions on the continu-
ity modulus of b and F to insure this uniqueness. This type of conditions
appears basically in [BK] where the authors gave optimal conditions for
uniqueness of entropy solution for the first order problem.
In this paper, we achieve the study of the problem (CP) by giving a
result for continuous dependence of the entropy solution. The contribution
of this work consists of considering (CP )(b, F, f, u0) on IRn instead of a
bounded domain. This is done by considering the same type of conditions
as in [MT3] and [BK]. Our main result is based on the theory and theorems
we have developed and proved in [MT1], [MT2] and [MT3].
In section 2 we recall the definitions of the types of solutions we use
(weak solution, entropy solution) and the main result presented in [MT3].
In section 3, we state and prove the main result (Theorem 6).
– 590 –
Continuous dependence of the entropy solution of general parabolic equation
2. Preliminaries
First of all we introduce the definitions of solution types used below.
Definition 2.1 (Weak solution of (CP )(b, F, f, u0)). — Let u0 and f
be such that (H1) is fulfilled. A weak solution of (CP ) is a function
u ∈ L∞(Q) such that:
ut ∈ L2((0, T ), H−1loc (IRN )) + L1((0, T ), L∞(IRN )), (2.1)
b(u) ∈ L2((0, T ), H1loc(IRN )) (2.2)
ut − ∆b(u) + divF (u) = f in D′(Q)
and u(0, x) = u0 in IRN.
The last condition is taken in the sense that∫ T
0
< ut, ξ > dt = −
∫
Q
u ξt dx dt−
∫
IRN
u0 ξ(0) dx, (2.3)
for any ξ ∈ L2((0, T );D(IRN )) ∩W 1,1((0, T );L∞(IRN )) so that ξ(T ) = 0
and <,> represents the duality product between H−1(IRN ) and H1(IRN).
It is well known, that we do not have uniqueness of weak solution in
general. In order to get uniqueness, we introduce entropy solution following
the notion of entropy introduced by S.N. Kruzkhov for conservation law
[KA], [C1], [C2] and their references
Definition 2.2 (Entropy solution of (CP ) (b, F, f, u0)). — Let u0 and
f verifying (H1). An entropy solution u of (CP ) (b, F, f, u0) is a weak so-
lution of (CP ) such that:∫
Q
H0(u− s){∇b(u)∇ξ − (F (u) − F (s))∇ξ − (u− s)ξt − fξ} dx dt
−
∫
IRN
(u0 − s)+ξ(0) dx  0
(2.4)
and∫
Q
H0(s− u){∇b(u)∇ξ − (F (u) − F (s))∇ξ − (u− s)ξt − fξ} dx dt
+
∫
IRN
(s− u0)+ξ(0) dx  0
(2.5)
for any s ∈ IR and ξ ∈ D(Q), ξ  0.
– 591 –
Mohamed Maliki
Let recall at first some important results from [MT3]. The first one is the
technical Lemma (cf Theorem 17 in [MT3]), it is the main tool in the proof
of the uniqueness of the entropy solution of (CP), in fact in this lemma we
can see the origin of conditions on the continuity modulus of b and F .
The second result is existence and uniqueness of the entropy solution of
(CP) (cf Theorem 14, Theorem 15 and Theorem 16 in [MT3]).
On the other hand, if u0 ∈ L1(IRN ) and f ∈ L1(Q), by the semi-group
theory, one knows that we have the existence and the uniqueness of the
mild solution of (CP) which is an entropy solution, moreover it depends
continuously in the data u0, f , b and F (cf Theorem 9, Theorem 10 in
[MT3]) (cf also [C1], [BCP]and their references).
Theorem 2.3 (The technical Lemma). — Let ω be non negative func-
tion


lim
ε→0
ω(ε)
ε
= +∞; lim
ε→0
ω(ε) = 0, there is l such that 2 < l  N
lim inf
ε→0
ωl(ε)
εl−1
< +∞ if N > 2, and lim inf
ε→0
ω2(ε)
ε
= 0 if N = 2,
(H2)
Let h ∈ L1loc(Q) such that h+ = max(h, 0) ∈ L1(Q). Let W0 ∈ L1(IRN ),
W ∈ L1loc(Q), W  0, W ∈ L∞(Q), and let λ > 0 so that:
∫ ∫
Q
W
∂ξ
∂t
+
l∑
j=1
(W + ε)
ω(ε)
ε
[| ∂
2ξ
∂xj∂xj
| + | ∂ξ
∂xj
|]
+
N∑
j=l+1
λW (| ∂
2ξ
∂xj∂xj
| + | ∂ξ
∂xj
|) + h ξ dx dt  0,
(2.6)
for any ε > 0 et ξ ∈ D(Q), ξ  0.
Set also that (W (t, .) − W0)+ −→ 0 in L1loc(IRN ), when t −→ 0
essentially; then h ∈ L1(Q), W ∈ L∞(0, T, L1(IRN )) and
∫
IRN
W (τ, x) dx 
∫
IRN
W0(x) dx+
∫
Qτ
h dx dt, for a.e. τ ∈ (0, T ),
with Qτ =]0, τ [×IRN .
– 592 –
Continuous dependence of the entropy solution of general parabolic equation
Remark 1. —
1. Theorem 2.3 is a particular case of Theorem 17 in [MT3] with ωj =
ηj = ω for j = 1...l.
2. Let g be a continuous function, we denote by ω a non negative con-
tinuous function which is non-decreasing such that ω(0) = 0 and
|g(x) − g(y)|  ω(x− y).
In general we have lim
ε→0
ω(ε)
ε
= +∞. In the particular case where
g is Lipschitz continuous, we have ω(ε)ε  λ where λ is a positive
constant.
Theorem 2.4 (Existence theorem) (cf. Theorem 14 in [MT3]). — For
all u0 and f verifying (H1). There exists an entropy solution of the problem
(CP )(b, F, f, uo).
Remark 2. — Proceeding if necessary to coordinate change, we can sup-
pose that:
• for i = 1...l the functions Fi are continuous but not Lipschitz contin-
uous.
• For i = l+1...N the functions Fi are Lipschitz continuous, we denote
by λ a constant such that |Fi(x) − Fi(y)|  λ|x− y|.
• We may suppose that l > 2, we replace if necessary λ by λ̃ = λ+ε
−1
N+1 .
• In the case where the function b is not Lipschitz continuous, we may
suppose l = N .
• For i = 1...l we denote by ω the continuity modulus of Fi and b .
Theorem 2.5 (Comparison principle). — Let (u01, f1), (u02, f2)
verifying (H1), and let u1, u2 be entropy solution with respect to (CP )
(b, F, f1, u01)), and (CP ) (b, F, f2, u02)). If (H2) holds, then:∫
IRN
(u1(t)−u2(t))+ dx 
∫
IRN
(u01−u02)+ dx+
∫
Qt
ν(f1−f2) dx dt. (2.7)
for ν ∈ H(u1 − u2) a.e so that:
‖u1(t) − u2(t)‖L1(IRN )  ‖u01 − u02‖L1(IRN ) +
∫ t
0
‖f1 − f2‖L1(IRN ) ds;
which gives uniqueness of entropy solution.
– 593 –
Mohamed Maliki
3. The main result
Our main result is:
Theorem 3.1. — Let be (Fn, bn, fn, u0,n) ∈ C(IR)× Cm(IR)× L1loc(Q)×
L∞(IR), given functions such that:


Fin → Fi for i=1...N bn → b in C(IR)
(fn, u0,n) converges to (f, u0) in L1loc(Q) × L1loc(IRN )
‖u0,n‖L∞(IRN ) +
∫ T
0
‖fn(t)‖L∞(IRN ) dt  C,
(3.1)
and we suppose that for n large enough we have


|Fin(x) − Fin(y)|  ω(x− y) for i = 1...l,
|Fin(x) − Fin(y)|  λ|x− y| for i = l + 1...N,
|bn(x) − bn(y)|  ω(x− y)
and (H2) hold.
(H3)
then un converges to u in C([0, T ], L1loc(IRN )), where un, u are respec-
tively the entropy solutions of (CP )(Fn, bn, fn, u0,n) and (CP )(F, b, f, u0).
Proof. — Let X be the set of (f̂ , û0) ∈ L1(Q) × L1(IRN ), such that
(H1) is satisfied and
‖û0‖L∞(IRN ) +
∫ T
0
‖f̂(t)‖
L∞(IRN ) dt  C
Let K be a compact in IRN , and δ ∈ IR+, there exist (f̂ , û0) ∈ X such
that ∫
K
|u0 − û0| +
∫ T
0
∫
K
|f − f̂ | dx ds < δ
3
; (3.2)
furthermore, since (f̂ , û0) is in X, by the semi group theory in L1 the Cauchy
problem (CP )(F, b, f̂ , û0), has a unique mild solution û (cf [BCP], [MT2],
[MT3]), which depends continuously on the data (F, b, f̂ , û0), and one has
sup
t∈[0,T ]
∫
K
|u(t) − û(t)| dx < δ
3
, (3.3)
where u is the unique entropy solution of the Cauchy problem (CP )(F, b, f, u0).
– 594 –
Continuous dependence of the entropy solution of general parabolic equation
Set ûn the entropy solution of the Cauchy problem (CP )(Fn, bn, f̂ , û0).
by the same arguments as before one has
lim
n→∞
sup
t∈[0,T ]
∫
IRN
|ûn(t) − û(t)| dx = 0. (3.4)
Moreover∫
K
|un(t) − u(t)| dx 
∫
K
|un(t) − ûn(t)| dx
+
∫
K
|ûn(t) − û(t)| dx+
∫
K
|û(t) − u(t)| dx.
(3.5)
By inequality (3.3), (3.4), and (3.5) a sufficient condition to get the result
of theorem 7 which can be written as:
lim
n→∞
sup
t∈[0,T [
∫
K
|un(t) − u(t)| dx = 0,
is to prove that there exist n0 ∈ IN, such that for n  n0
sup
t∈[0,T ]
∫
K
|un(t) − ûn(t)| dx <
δ
3
.
By Theorem 16 in [MT3] for all ξ ∈ D(IR), ξ  0 et 0 < t  T the “kato”
inequality applied to (CP )(Fn, bn, fn, u0,n) and (CP )(Fn, bn, f̂n, û0,n) leads
to: ∫
IRN
|un(t) − ûn(t)| ξ dx 
∫
IRN
|u0,n − û0,n| ξ dx
+
N∑
j=1
∫
Q
|bn(un) − bn(ûn)||
∂2ξ
∂xj∂xj
|
+
N∑
j=1
∫
Q
|Fin(un) − Fin(ûn)||
∂ξ
∂xj
|
+
∫
IR
|fn − f̂ | ξ dx ds,
(3.6)
In order to use Theorem 2.3 we set
Wn(t) = |un(t) − ûn(t)|; W0,n = |u0,n − û0,n|; hn = |fn − f̂ |.
– 595 –
Mohamed Maliki
By using the two first inequalities in (H3), one obtain for n  n0∫
IRN
Wn(t) ξ dx 
∫
IRN
W0,n ξ dx+
∫
IR
hn ξ dx ds
+2
l∑
j=1
∫ t
0
∫
IRN
(Wn + ε)
ω(ε)
ε
(| ∂
2ξ
∂xj∂xj
| + | ∂ξ
∂xj
|) dx ds
+2
N∑
j=l+1
∫ t
0
∫
IRN
λWn(|
∂2ξ
∂xj∂xj
| + | ∂ξ
∂xj
|) dx ds,
(3.7)
withW0,n → 0 dans L1loc(IRN ), hn → 0 dans L1loc(Q), and ‖Wn‖L∞(Q)  C,
By the proof of Theorem 16 of [MT3] one has
1
2
sup
τ∈(0,T )
ess
∫
IRN
Wn(τ)ξ dx

∫
IRN
W0,nξ dx+
∫ τ
0
∫
IRN
hnξ dt dx + Λ(ε) (3.8)
where
1) ξ is a particular test function which tends to 1 when ε tends to 0.
2) 0  Λ(ε)  M ω
l(ε)
ε1−lηl−2
, M is a constant.
3) η is a parameter assigned to go to +∞.
From (3.8) one has that for n large enough:
1
2
sup
τ∈(0,T )
ess
∫
K
W (τ) ξ dx  δ
3
+
δ
3
+ Λ(ε) (3.9)
Then
If N > 2 we let η goes to +∞ and ε goes to 0 to get the result.
If N = 2 then l = 2 and Λ(ε)  Mω
2(ε)
ε so we let ε going to 0 to get
the result.
If N = 1 there is no condition on ω.
– 596 –
Continuous dependence of the entropy solution of general parabolic equation
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– 598 –


Continuous dependence of the entropy solution of general parabolic equation

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