Partial Differential Equations
On nonlinear diffusion problems with strong degeneracy
Comptes Rendus. Mathématique, Volume 343 (2006) no. 9, pp. 569-572.

In this Note, we study the ‘triply’ degenerate problem: b(v)tΔg(v)+divΦ(v)=f on Q:=(0,T)×Ω, b(v(0,))=b(v0) on Ω and g(v)=g(a) ‘on some part of the boundary’ (0,T)×Ω, in the case of continuous nonhomogenous and nonstationary boundary data a. The functions b,g are assumed to be continuous nondecreasing and to verify the normalisation condition b(0)=g(0)=0 and the range condition R(b+g)=R. Using monotonicity and penalization methods, we prove existence of a weak entropy solution in the spirit of F. Otto (1996).

Dans cette Note, on étudie le problème triplement dégénéré : b(v)tΔg(v)+divΦ(v)=f sur Q:=(0,T)×Ω, b(v(0,))=b(v0) dans Ω et g(v)=g(a) « sur une partie de la frontière » (0,T)×Ω, dans le cas d'une donnée a continue non homogène et non stationnaire sur le bord. Les fonctions b,g sont supposées être continues croissantes, vérifiant la condition de normalisation : b(0)=g(0)=0 et de surjectivité R(b+g)=R. En utilisant des méthodes de monotonie et de pénalisation, on prouve l'existence d'une solution entropique au sens de F. Otto (1996).

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Accepted:
Published online:
DOI: 10.1016/j.crma.2006.09.030
Ammar, Kaouther 1

1 Institut für Mathematik, TU Berlin, Strasse des 17 juni 135, 10625 Berlin, Germany
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Ammar, Kaouther. On nonlinear diffusion problems with strong degeneracy. Comptes Rendus. Mathématique, Volume 343 (2006) no. 9, pp. 569-572. doi : 10.1016/j.crma.2006.09.030. http://www.numdam.org/articles/10.1016/j.crma.2006.09.030/

[1] K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations (2006), in press

[2] Bardos, C.; LeRoux, A.Y.; Nedelec, J.C. First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, Volume 4 (1979) no. 9, pp. 1017-1034

[3] Carrillo, J. Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech. Math., Volume 147 (1999), pp. 269-361

[4] Mascia, C.; Porretta, A.; Terracina, A. Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Rational Mech. Anal., Volume 163 (2002), pp. 87-124

[5] Michel, A.; Vovelle, J. Entropy formulation for a parabolic degenerate problem with general Dirichlet boundary conditions and application to the convergence of FV methods, SIAM J. Numer. Anal., Volume 41 (2003) no. 6, pp. 2262-2293

[6] Otto, F. Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris, Volume 322 (1996), pp. 729-734

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