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

Jean-Michel Bony

Kaouther Ammar ¹

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

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

Abstract (Original language)
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).

Article information
Cite this article

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

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

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References
Cited by

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