Convergence in $Clbrack0,T brack;L^2Omega$ of weak solutions to perturbed doubly degenerate parabolic equationsReport as inadecuate




Convergence in $Clbrack0,T brack;L^2Omega$ of weak solutions to perturbed doubly degenerate parabolic equations - Download this document for free, or read online. Document in PDF available to download.

1 Monash University 2 LAMA - Laboratoire d-Analyse et de Mathématiques Appliquées

Abstract : We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic p-Laplace equation. We show that, up to a subsequence, weak solutions of the perturbed problem converge uniformly-in-time to weak solutions of the original problem as the perturbed data approach the original data. We do not assume uniqueness or additional regularity of the solution. However, when uniqueness is known, our result demonstrates that the weak solution is uniformly temporally stable to perturbations of the data. Beginning with a proof of temporally-uniform, spatially-weak convergence, we strengthen the latter by relating the unknown to an underlying convex structure that emerges naturally from energy estimates on the solution. The double degeneracy — shown to be equivalent to a maximal monotone operator framework — is handled with techniques inspired by a classical monotonicity argument and a simple variant of the compensated compactness phenomenon.

Keywords : uniform temporal convergence Minty-Browder monotonicity technique degenerate parabolic equation Leray-Lions operator maximal monotone operator Richards equation Stefan problem





Author: Jerome Droniou - Robert Eymard - Kyle Talbot -

Source: https://hal.archives-ouvertes.fr/



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