Renormalized solution for nonlinear degenerate problems in the whole space

Mohamed Maliki; Adama Ouedraogo

doi:10.5802/afst.1194

Mohamed Maliki ¹; Adama Ouedraogo ²

¹ Department of Mathematics BP 146, Hassan II University Mohammedia (Morocco)
² Department of Mathematics 03 BP 7021 University of Ouagadougou 03 (Burkina Faso)

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 3, pp. 597-611.

Abstract
Abstract

We consider the general degenerate parabolic equation :

u_{t} - Δ b (u) + d i v \tilde{F} (u) = f in Q =] 0, T [\times ℝ^{N}, T > 0 .

We suppose that the flux $\tilde{F}$ is continuous, $b$ is nondecreasing continuous and both functions are not necessarily Lipschitz. We prove the existence of the renormalized solution of the associated Cauchy problem for $L^{1}$ initial data and source term. We establish the uniqueness of this type of solution under a structure condition $\tilde{F} (r) = F (b (r))$ and an assumption on the modulus of continuity of $b$ . The novelty of this work is that $Ω = ℝ^{N}$ , $u_{0}$ , $f \in L^{1}$ , $b$ , $\tilde{F}$ are not Lipschitz functions and the techniques are different from those developed in the previous works.

Nous considérons l’équation parabolique dégénérée général :

u_{t} - Δ b (u) + d i v \tilde{F} (u) = f dans Q =] 0, T [\times ℝ^{N}, T > 0 .

Nous supposons que le flux $\tilde{F}$ est continu, $b$ est continue et croissante au sens large et les deux fonctions ne sont pas nécessairement lipschitziennes. Nous prouvons l’existence de solution renormalisée du problème de Cauchy associé à cette équation avec des données (terme source et condition initiale) dans $L^{1}$ . Nous établissons l’unicité de cette solution sous une condition dite de structure du type $\tilde{F} (r) = F (b (r))$ et sous une hypothèse sur le module de continuité de $b$ . La nouveauté dans le travail vient du fait que $Ω = ℝ^{N}$ , $u_{0}$ , $f \in L^{1}$ , $b$ , $\tilde{F}$ ne sont pas des fonctions nécessairement lipschitziennes et les techniques sont différentes de celles développées dans les travaux antérieurs.

MR Zbl

DOI: 10.5802/afst.1194

Author's affiliations:

Mohamed Maliki ¹; Adama Ouedraogo ²

¹ Department of Mathematics BP 146, Hassan II University Mohammedia (Morocco)
² Department of Mathematics 03 BP 7021 University of Ouagadougou 03 (Burkina Faso)

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     author = {Mohamed Maliki and Adama Ouedraogo},
     title = {Renormalized solution for nonlinear degenerate problems in the whole space},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {597--611},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 17},
     number = {3},
     year = {2008},
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Mohamed Maliki; Adama Ouedraogo. Renormalized solution for nonlinear degenerate problems in the whole space. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 3, pp. 597-611. doi : 10.5802/afst.1194. https://afst.centre-mersenne.org/articles/10.5802/afst.1194/

References
Cited by

[AW] Ammar (K.), Wittbold (P.).— Existence of renormalized solution of degnerate elliptic-parbolic problems Proc. Royal Soc. Edinb.113A, p. 477-496, (2003). | MR | Zbl

[AI] Andreianov (B.P.), Igbida (N.).— Uniqueness for Nonlinear degenerate diffusion-convection problem, to appear in J. Diff. Equat. | Zbl

[ABK] Andreianov (B.P.), Bénilan (Ph.), Kruzhkov (S.N.).— $L^{1}$ theory of scalar conservation law with continuous flux function. J. Funct. Anal, 171 p. 415-33 (2000). | Zbl

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

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

[BBGPV] Bénilan (Ph.), Boccardo (L.), Gariepy (B.), Pierre (M.), Vazquez (J.L.).— An $L^{1}$ theory of exsitence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scul. Norm. Sup. 22(2) p. 241-273, (1995). | Numdam | MR | Zbl

[BCBT] Butos (M.C.), Concha (F.), Bürger (R.), Tory (E.M.).— Sedimentation and thickenning : phenomenological foundation and mathematical theory, Kluwer Academic, Dordrecht, (1999). | MR | Zbl

[BM] Blanchart (D.), Murat (F.).— Renormalized solutions of nonlinear parabolic problems with $L^{1}$ data : existence and uniqueness Proc.royal Soc. Edinb A 127, p. 1137-1152, (1997). | MR | Zbl

[BR] Blanchard (D.), Redouane (H.).— Solutions renormalisées d’équations paraboliques à deux non linéaritées. C.R.A.S. Paris 319, p. 831-847, (1994). | Zbl

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

[BK] Bénilan (Ph.), Kruzhkov (S.N.).— Quasilinear first order equations with continuous non linearities. Russian Acad. Sci. Dokl. Math. Vol 50 $N^{\circ} 3$ , p. 391-396 (1995). | MR | Zbl

[BT1] Bénilan (Ph.), Touré (H.).— Sur l’équation générala $u_{t} = ϕ {(u)}_{x x} - ψ {(u)}_{x} + v,$ C.R. Acad. Sc. Paris, serie 1, 299, 18 (1984). | Zbl

[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 Vol. 168, (1995).

[BT3] Bénilan (Ph.), Touré (H.).— Sur l’équation générale $u_{t} = a {(., u, ϕ {(., u)}_{x})}_{x}$ dans $L^{1}$ $I I$ . Le problème d’évolution, Ann. Inst. Henri Poincaré, vol. 12, 6, p. 727-761 (1995). | Numdam | Zbl

[BW] Bénilan (Ph.), Wittbold (P.).— On mild and weak solution of elliptic-Parabolic Problems. Adv. in Diff. Equat. Vol. 1 (6) p. 1053-1072 (1996). | MR | Zbl

[C1] Carrillo (J.).— On the uniquness of the solution of the evolution DAM problem, Nonlinear Analysis, Vol 22, $N^{\circ} 5$ , p. 573-607 (1999). | MR | Zbl

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

[C3] Carrillo (J.).— Unicité des solutions du type Kruskov pour des problèmes elliptiques avec des termes de transport non linéaires C. R. Acad. Sc. Paris, t 33, Serie I, $N^{\circ} 5$ , (1986). | MR | Zbl

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

[CJ] Chavent (G.), Jaffre (J.).— Mathematical models and finite elements for reservoir simulation, North Holland, Amsterdam, (1986). | Zbl

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

[GT] Gagneux (G.).— Tort (M.M.), Unicité des solutions faibles d’équations de diffusion convection, C. R. Acad. SC. Paris, t 318, Série I, p. 919-924 (1994). | MR | Zbl

[HV] Hudjaev (S.N.), Vol’pert (A.I.).— Cauchy’s problem for degenerate second order quasilinear parabolic equation, Math. USSR-Sbornik, Vol 7, $N^{\circ} 3$ , p. 365-387 (1969). | Zbl

[IU1] Igbida (N.), Urbano (J.M.).— Uniqueness for degenerate problems, NoDEA 10, p. 287-307 (2003). | MR | Zbl

[IU2] Igbida (N.), Urbano (J.M.).— Continuity results for certain nonlinear parabolic PDEs, Preprint LAMFA, Université de Picadie Jules Vernes.

[IW] Igbida (N.), Wittbold (P.).— Renormalized solution for stephan type problem : Existence and Uniqueness, Preprint LAMFA, Université de Picadie Jules Vernes.

[KA] Kruzhkov (S.N.), Panov (E. Yu.).— Conservative quasilinear first order law in the class of locally sommable functins, Dokl. Akad. Nauk. S.S.S.R. 220, 1 p. 233-26 ; english traduction in soviet Math. Dokl. 16 (1985).

[L] Landes (R.).— On the existence of weak solutions for quasilinear parabolic initial boundary-value problems.Proc. Royal Soc. Edinb. 89A:217-237, (1981). | MR | Zbl

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

[M] Maliki (M.).— Continuous dependence of the entropy solution of general parabolic equation, Ann. Fac. Sci. Toulouse. Vol.XV,n¡3, p. 589-598 (2006). | Numdam | MR | Zbl

[MK] Bendahmane (M.), Karlsen (K.H.).— Renormalized solutions for quasilinear anisotropic degenerate parabolic equations, Siam J. Marth.Anal. Vol.36, N0.2, pp.405-422, (2004). | MR | Zbl

[MT1] Maliki (M.), Touré (H.).— Solution généralisée locale d’une équation parabolique quasi linéaire dégénérée du second ordre. Ann. Fac. Sci. Toulouse. Vol. VII 1, (1998) 113-133. | Numdam | MR | Zbl

[MT2] Maliki (M.), Touré ( H.).— Dépendence continue de solutions généralisées locales, Ann. Fac. Sci. Toulouse. Vol.X 4, (2001) 701-711. | Numdam | MR | Zbl

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

[KP] Kruzhkov (S.N.), Panov E.Yu..— Conservative quasilinear first order laws with an infinite domain of dependence on the initial data, Soviet. Math. Dokl. Vol. 42, 2, p. 316-321 (1991). | MR | Zbl

[O] Otto (P.).— $L^{1}$ contraction and uniqueness for quasilinear elliptic-parobolic equations J. Diff. Eqa. 131, p. 20-38 (1996). | MR | Zbl

[YJ] Yin (J.).— On the uniqueness and stability of BV solutions for nonlinear diffusion equations, Comm. Part. Diff. Equat. 15, 12, p. 1671-1683 (1990). | MR | Zbl

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