Convergence in $C(\lbrack0,T\rbrack;L^2(\Omega))$ of weak solutions to perturbed doubly degenerate parabolic equations

TL;DR

This paper proves that solutions to certain nonlinear degenerate parabolic equations remain stable under data perturbations, with convergence of solutions demonstrated without requiring uniqueness or extra regularity assumptions.

Contribution

It establishes uniform-in-time convergence of weak solutions for a broad class of degenerate parabolic problems, including the Richards and Stefan equations, under data perturbations.

Findings

Weak solutions converge uniformly in time as data perturbations vanish

The approach handles double degeneracy via maximal monotone operator framework

Results hold without assuming solution uniqueness or additional regularity

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.