Error estimate for classical solutions to the heat equation in a moving thin domain and its limit equation

TL;DR

This paper provides an error estimate for classical solutions to the heat equation in a moving thin domain and analyzes its limit equation on the hypersurface, using uniform a priori estimates and asymptotic expansions.

Contribution

It introduces a novel error estimate in the sup-norm for solutions in moving thin domains and connects it rigorously to the limit equation on the hypersurface.

Findings

Established a uniform a priori estimate for the thin domain problem

Constructed an approximate solution from the limit equation

Derived an error bound in the sup-norm for the classical solutions

Abstract

We consider the Neumann type problem of the heat equation in a moving thin domain around a given closed moving hypersurface. The main result of this paper is an error estimate in the sup-norm for classical solutions to the thin domain problem and a limit equation on the moving hypersurface which appears in the thin-film limit of the heat equation. To prove the error estimate, we show a uniform a priori estimate for a classical solution to the thin domain problem based on the maximum principle. Moreover, we construct a suitable approximate solution to the thin domain problem from a classical solution to the limit equation based on an asymptotic expansion of the thin domain problem and apply the uniform a priori estimate to the difference of the approximate solution and a classical solution to the thin domain problem.