Continuous dependence of the entropy solution of general parabolic equation

Mohamed Maliki

doi:10.5802/afst.1130

Mohamed Maliki ¹

¹ Université Hassan II, F.S.T. Mohammedia, B.P. 146 Mohammedia, Maroc.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, pp. 589-598.

Abstract
Abstract

We consider the general parabolic equation :

$u_{t} - Δ b (u) + d i v F (u) = f$ in $Q =] 0, T [\times ℝ^{N}, T > 0$ with $u_{0} \in L^{\infty} (ℝ^{N}),$ $for a . e t \in] 0, T [, f (t) \in L^{\infty} (ℝ^{N})$ and $\int_{0}^{T} {∥f (t)∥}_{L^{\infty} (ℝ^{N})} d t < \infty .$

We prove the continuous dependence of the entropy solution with respect to $F,$ $b,$ $f$ and the initial data $u_{0}$ 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 $ℝ^{n}$ instead of a bounded domain and considering a bounded data instead of integrable data.

On considère l’équation parabolique générale :

$u_{t} - Δ b (u) + d i v F (u) = f dans Q =] 0, T [\times ℝ^{N}, T > 0$ avec $u_{0} \in L^{\infty} (ℝ^{N}), f \in L_{L o c}^{1} (Q)$ pour $p . p t \in] 0, T [f (t) \in L^{\infty} (ℝ^{N})$ ,

et $\int_{0}^{T} {∥f (t)∥}_{L^{\infty} (ℝ^{N})} d t < \infty .$

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 $u_{0}$ . 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]. Ensuite on montre le résultat principal du travail en utilisant le lemme abstrait (Théorème 2.3). La contribution du travail est de traiter le problème dans $ℝ^{N}$ ainsi que de considérer des données bornées au lieu des données intégrables utilisées dans la littérature.

MR Zbl

DOI: 10.5802/afst.1130

Author's affiliations:

Mohamed Maliki ¹

¹ Université Hassan II, F.S.T. Mohammedia, B.P. 146 Mohammedia, Maroc.

@article{AFST_2006_6_15_3_589_0,
     author = {Mohamed Maliki},
     title = {Continuous dependence of the entropy solution of general parabolic equation},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {589--598},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 15},
     number = {3},
     year = {2006},
     doi = {10.5802/afst.1130},
     mrnumber = {2246415},
     zbl = {1122.35012},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1130/}
}

TY  - JOUR
AU  - Mohamed Maliki
TI  - Continuous dependence of the entropy solution of general parabolic equation
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2006
SP  - 589
EP  - 598
VL  - 15
IS  - 3
PB  - Université Paul Sabatier, Institut de Mathématiques
PP  - Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1130/
DO  - 10.5802/afst.1130
LA  - en
ID  - AFST_2006_6_15_3_589_0
ER  -

%0 Journal Article
%A Mohamed Maliki
%T Continuous dependence of the entropy solution of general parabolic equation
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2006
%P 589-598
%V 15
%N 3
%I Université Paul Sabatier, Institut de Mathématiques
%C Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1130/
%R 10.5802/afst.1130
%G en
%F AFST_2006_6_15_3_589_0

Mohamed Maliki. Continuous dependence of the entropy solution of general parabolic equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, pp. 589-598. doi : 10.5802/afst.1130. https://afst.centre-mersenne.org/articles/10.5802/afst.1130/

References
Cited by

[ABK] B. P. Andreianov; Ph. Bénilan; S. N. Kruskhov $L^{1}$ theory of scalar conservation law with continuous flux function, J. Funct. Anal., Volume 171 (2000), pp. 15-33 | MR | Zbl

[AL] H. W. Alt; S. Luckhauss Quasi-linear elliptic-parabolic differential equations, Math.Z., Volume 183 (1983), pp. 311-341 | MR | Zbl

[BCP] Ph. Bénilan; M. G. Crandall; A. Pazy Evolution Equation governed by Accretive operators (book to appear)

[BG] Ph. Bénilan; B. Gariepy Strong solution $L^{1}$ of degenerate parabolic equation, J. of Diff. Equat., Volume 119 (1995), pp. 473-502 | MR | Zbl

[BK] Ph. Bénilan; S. N. Kruskhov Quasilinear first order equations with continuous non linearities, Russian Acad. Sci. Dokl. Math., Volume 50 (1995) no. 3, pp. 391-396 | MR | Zbl

[BT1] Ph. Bénilan; H. Touré Sur l’équation générale $u_{t} = ϕ {(u)}_{x x} - ψ {(u)}_{x} + v$ , C.R. Acad. Sc. Paris, série 1, Volume 299 (1984), pp. 919-922 | MR | Zbl

[BT2] Ph. Bénilan; H. Touré Sur l’équation générale $u_{t} = a {(., u, ϕ {(., u)}_{x})}_{x}$ dans $L^{1}$ I. Etude du problème stationnaire, in Evolution equations, Lecture Notes Pure and Appl. Math., Volume 168 (1995) | Zbl

[BT3] Ph. Bénilan; H. Touré Sur l’équation générale $u_{t} = a {(., u, ϕ {(., u)}_{x})}_{x}$ dans $L^{1}$ II. Le problème d’évolution, Ann. Inst. Henri Poincaré, Volume 12 (1995) no. 6, pp. 727-761 | Numdam | MR | Zbl

[BW] Ph. Bénilan; P. Wittbold On mild and weak solution of elliptic-Parabolic Problems, Adv. in Diff. Equat., Volume 1 (1996) no. 6, pp. 1053-1072 | MR | Zbl

[C1] J. Carrillo On the uniquness of the solution of the evolution DAM problem, Nonlinear Analysis, Volume 22 (1999) no. 5, pp. 573-607 | MR | Zbl

[C2] J. Carrillo Entropy solutions for nonlinear degenerate problems, Arch. Ratio. Mech. Anal., Volume 147 (1999), pp. 269-361 | MR | Zbl

[C3] J. Carrillo Unicité des solutions du type Kruskhov pour des problèmes elliptiques avec des termes de transport non linéaires, C. R. Acad. Sc. Paris, Série I, Volume 33 (1986) no. 5 | MR | Zbl

[CW] J. Carrillo; P. Wittbold Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Diff. Equation, Volume 156 (1999), pp. 93-121 | MR | Zbl

[DT] J. I. Diaz; F. Thelin On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM. J. Math. Anal., Volume 25 (1994), pp. 1085-1111 | MR | Zbl

[GT] G. Gagneux; M. M. Tort Unicité des solutions faibles d’équations de diffusion convection, C. R. Acad. SC. Paris, Série I, Volume 318 (1994) | MR | Zbl

[KA] S. N. Kruskhov; E. Yu. Panov Conservative quasilinear first order law in the class of locally sommable functins, Dokl. Akad. Nauk. S.S.S.R., Volume 220 (1985) no. 1, p. 233-26 (english traduction in soviet Math. Dokl. 16)

[KP] S. N. Kruskhov; E. Yu. Panov Conservative quasilinear first order laws with an infinite domain of dependence on the initial data, Soviet. Math. Dokl., Volume 42 (1991) no. 2, pp. 316-321 | MR | Zbl

[LSU] O. A. Ladyzenskaja; V. A. Solonnikov; N. N. Ural’ceva Linear and quasilinear equations of parabolic type, Transl. of Math. Monographs, Volume 23 (1968) | MR | Zbl

[MT1] M. Maliki; H. Touré Solution généralisée locale d’une équation parabolique quasi linéaire dégénérée du second ordre, Ann. Fac. Sci. Toulouse, Volume VII (1998) no. 1, pp. 113-133 | EuDML | Numdam | MR | Zbl

[MT2] M. Maliki; H. Touré Dépendence continue de solutions généralisées locales, Ann. Fac. Sci. Toulouse, Volume X (2001) no. 4, pp. 701-711 | EuDML | Numdam | MR | Zbl

[MT3] M. Maliki; H. Touré Uniqueness entropy solutions for nonlinear degenerate parabolic problem, Journal of Evolution equation, Volume 3 (2003) no. 4, pp. 603-622 | MR | Zbl

[YJ] J. Yin On the uniqueness and stability of BV solutions for nonlinear diffusion equations, Comm. Part. Diff. Equat., Volume 15 (1990) no. 12, pp. 1671-1683 | MR | Zbl

Cited by Sources: