### 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 language | English |
---|---|

Pages (from-to) | 7821-7860 |

Number of pages | 40 |

Journal | Journal of Differential Equations |

Volume | 260 |

Issue number | 11 |

DOIs | |

Publication status | Published - 5 Jun 2016 |

### Keywords

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

### Cite this

^{2}(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations.

*Journal of Differential Equations*,

*260*(11), 7821-7860. https://doi.org/10.1016/j.jde.2016.02.004

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^{2}(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations',

*Journal of Differential Equations*, vol. 260, no. 11, pp. 7821-7860. https://doi.org/10.1016/j.jde.2016.02.004

**Convergence in C([0,T];L ^{2}(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations.** / Droniou, Jerome; Eymard, Robert; Talbot, Kyle S.

Research output: Contribution to journal › Article › Research › peer-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

UR - http://www.scopus.com/inward/record.url?scp=84964409608&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2016.02.004

DO - 10.1016/j.jde.2016.02.004

M3 - Article

VL - 260

SP - 7821

EP - 7860

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 11

ER -

^{2}(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations. Journal of Differential Equations. 2016 Jun 5;260(11):7821-7860. https://doi.org/10.1016/j.jde.2016.02.004