Continuous dependence of the entropy solution of general parabolic equation
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 3, pp. 589-598.

On considère l’équation parabolique générale :

u t -Δb(u)+divF(u)=fdansQ=]0,T[× N ,T>0 avec u 0 L ( N ),fL Loc 1 (Q) pour p.pt]0,T[f(t)L ( N ),

et 0 T f(t) L ( N ) 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 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 N ainsi que de considérer des données bornées au lieu des données intégrables utilisées dans la littérature.

We consider the general parabolic equation :

u t -Δb(u)+divF(u)=f in Q=]0,T[× N ,T>0 with u 0 L ( N ), fora.et]0,T[,f(t)L ( N ) and 0 T f(t) L ( N ) dt<.

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 n instead of a bounded domain and considering a bounded data instead of integrable data.

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     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},
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Maliki, Mohamed. Continuous dependence of the entropy solution of general parabolic equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 3, pp. 589-598. doi : 10.5802/afst.1130. http://www.numdam.org/articles/10.5802/afst.1130/

[ABK] Andreianov, B. P.; Bénilan, Ph.; Kruskhov, S. N. L 1 theory of scalar conservation law with continuous flux function, J. Funct. Anal., Volume 171 (2000), pp. 15-33 | MR 1742856 | Zbl 0944.35048

[AL] Alt, H. W.; Luckhauss, S. Quasi-linear elliptic-parabolic differential equations, Math.Z., Volume 183 (1983), pp. 311-341 | MR 706391 | Zbl 0497.35049

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

[BG] Bénilan, Ph.; Gariepy, B. Strong solution L 1 of degenerate parabolic equation, J. of Diff. Equat., Volume 119 (1995), pp. 473-502 | MR 1340548 | Zbl 0828.35050

[BK] Bénilan, Ph.; Kruskhov, S. N. Quasilinear first order equations with continuous non linearities, Russian Acad. Sci. Dokl. Math., Volume 50 (1995) no. 3, pp. 391-396 | MR 1316937 | Zbl 0880.35027

[BT1] Bénilan, Ph.; Touré, H. Sur l’équation générale u t =ϕ(u) xx -ψ(u) x +v, C.R. Acad. Sc. Paris, série 1, Volume 299 (1984), pp. 919-922 | MR 774668 | Zbl 0586.35016

[BT2] Bénilan, Ph.; Touré, H. 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 0820.34011

[BT3] Bénilan, Ph.; Touré, H. 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 1360542 | Zbl 0839.35068

[BW] Bénilan, Ph.; Wittbold, P. On mild and weak solution of elliptic-Parabolic Problems, Adv. in Diff. Equat., Volume 1 (1996) no. 6, pp. 1053-1072 | MR 1409899 | Zbl 0858.35064

[C1] Carrillo, J. On the uniquness of the solution of the evolution DAM problem, Nonlinear Analysis, Volume 22 (1999) no. 5, pp. 573-607 | MR 1266545 | Zbl 0810.76086

[C2] Carrillo, J. Entropy solutions for nonlinear degenerate problems, Arch. Ratio. Mech. Anal., Volume 147 (1999), pp. 269-361 | MR 1709116 | Zbl 0935.35056

[C3] Carrillo, J. 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 854731 | Zbl 0623.35030

[CW] Carrillo, J.; Wittbold, P. Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Diff. Equation, Volume 156 (1999), pp. 93-121 | MR 1701806 | Zbl 0932.35129

[DT] Diaz, J. I.; Thelin, F. On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM. J. Math. Anal., Volume 25 (1994), pp. 1085-1111 | MR 1278892 | Zbl 0808.35066

[GT] Gagneux, G.; Tort, M. M. Unicité des solutions faibles d’équations de diffusion convection, C. R. Acad. SC. Paris, Série I, Volume 318 (1994) | MR 1278152 | Zbl 0826.35057

[KA] Kruskhov, S. N.; Panov, E. Yu. 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] Kruskhov, S. N.; Panov, E. Yu. 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 1118483 | Zbl 0789.35039

[LSU] Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N. Linear and quasilinear equations of parabolic type, Transl. of Math. Monographs, Volume 23 (1968) | MR 241822 | Zbl 0174.15403

[MT1] Maliki, M.; Touré, H. 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 73442 | Numdam | MR 1658456 | Zbl 0914.35068

[MT2] Maliki, M.; Touré, H. Dépendence continue de solutions généralisées locales, Ann. Fac. Sci. Toulouse, Volume X (2001) no. 4, pp. 701-711 | EuDML 73564 | Numdam | MR 1944257 | Zbl 1029.35027

[MT3] Maliki, M.; Touré, H. Uniqueness entropy solutions for nonlinear degenerate parabolic problem, Journal of Evolution equation, Volume 3 (2003) no. 4, pp. 603-622 | MR 2058053 | Zbl 1052.35106

[YJ] Yin, J. 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 1080617 | Zbl 0726.35063

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