Convergence in C([0,T];L²(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations

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• 作者：Jé ; rô ; me Droniou^a ; ^{jerome.droniou@monash.edu" class="auth_mail" title="E-mail the corresponding author} ; Robert Eymard^b ; ^{robert.eymard@u-pem.fr" class="auth_mail" title="E-mail the corresponding author} ; Kyle S. Talbot^a ; ^{kyle.talbot@monash.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35D30 ; 35K20 ; 35K55 ; 35K65 ; 35K92
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 June 2016
• 年：2016
• 卷：260
• 期：11
• 页码：7821-7860
• 全文大小：543 K

文摘

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 regularity. 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. 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.