On nonlinear diffusion problems with strong degeneracy

Kaouther Ammar

doi:10.1016/j.crma.2006.09.030

Partial Differential Equations

On nonlinear diffusion problems with strong degeneracy

Presented by: Jean-Michel Bony

Kaouther Ammar ¹

¹ Institut für Mathematik, TU Berlin, Strasse des 17 juni 135, 10625 Berlin, Germany

Comptes Rendus. Mathématique, Volume 343 (2006) no. 9, pp. 569-572.

Abstract
Résumé

In this Note, we study the ‘triply’ degenerate problem: $b {(v)}_{t} - Δ g (v) + div Φ (v) = f$ on $Q : = (0, T) \times Ω$ , $b (v (0, \cdot)) = b (v_{0})$ on Ω and $g (v) = g (a)$ ‘on some part of the boundary’ $(0, T) \times \partial Ω$ , 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 : $b {(v)}_{t} - Δ g (v) + div Φ (v) = f$ sur $Q : = (0, T) \times Ω$ , $b (v (0, \cdot)) = b (v_{0})$ dans Ω et $g (v) = g (a)$ « sur une partie de la frontière » $(0, T) \times \partial Ω$ , 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).

Received: 2006-06-29
Accepted: 2006-09-26
Published online: 2006-10-23

DOI: 10.1016/j.crma.2006.09.030

Author's affiliations:

Kaouther Ammar ¹

¹ Institut für Mathematik, TU Berlin, Strasse des 17 juni 135, 10625 Berlin, Germany

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     author = {Kaouther Ammar},
     title = {On nonlinear diffusion problems with strong degeneracy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {569--572},
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     number = {9},
     year = {2006},
     doi = {10.1016/j.crma.2006.09.030},
     language = {en},
}

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Kaouther Ammar. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.09.030/

References
Cited by

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

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

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

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