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

Jerome Droniou, Robert Eymard, Kyle S. Talbot

Research output: Contribution to journalArticleResearchpeer-review

7 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)7821-7860
Number of pages40
JournalJournal of Differential Equations
Volume260
Issue number11
DOIs
Publication statusPublished - 5 Jun 2016

Keywords

  • Degenerate parabolic equation
  • Leray–Lions operator
  • Maximal monotone operator
  • Richards equation
  • Stefan problem
  • Uniform temporal convergence

Cite this

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title = "Convergence in C([0,T];L2(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations",
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 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.",
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Convergence in C([0,T];L2(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations. / Droniou, Jerome; Eymard, Robert; Talbot, Kyle S.

In: Journal of Differential Equations, Vol. 260, No. 11, 05.06.2016, p. 7821-7860.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - Convergence in C([0,T];L2(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations

AU - Droniou, Jerome

AU - Eymard, Robert

AU - Talbot, Kyle S.

PY - 2016/6/5

Y1 - 2016/6/5

N2 - 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.

AB - 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.

KW - Degenerate parabolic equation

KW - Leray–Lions operator

KW - Maximal monotone operator

KW - Richards equation

KW - Stefan problem

KW - Uniform temporal convergence

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