Thin-film limit of the parabolic $p$-Laplace equation in a moving thin domain

TL;DR

This paper rigorously derives a limit problem for the parabolic p-Laplace equation in a moving thin domain as it collapses onto a hypersurface, revealing a new local mass conservation law on the surface.

Contribution

It introduces a novel limit problem combining a nonlinear PDE and algebraic equation on a moving hypersurface, extending understanding of thin domain limits.

Findings

Derived the limit problem as the thin domain shrinks to a hypersurface.

Characterized the limit function as a unique weak solution.

Revealed a new local mass conservation law on the moving hypersurface.

Abstract

We consider the parabolic $p$-Laplace equation with $p>2$ in a moving thin domain under a Neumann type boundary condition corresponding to the total mass conservation. When the moving thin domain shrinks to a given closed moving hypersurface as its thickness tends to zero, we rigorously derive a limit problem by showing the weak convergence of the weighted average of a weak solution to the thin-domain problem and characterizing the limit function as a unique weak solution to the limit problem. The limit problem obtained in this paper is a system of a nonlinear partial differential equation and an algebraic equation on the moving hypersurface. This seems to be somewhat strange, but we also find that the limit problem can be seen as a new kind of local mass conservation law on the moving hypersurface with a normal flux.