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

### Tatsu-Hiko Miura

Hirosaki University, Japan

## 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.

## Cite this article

Tatsu-Hiko Miura, Error estimate for classical solutions to the heat equation in a moving thin domain and its limit equation. Interfaces Free Bound. 25 (2023), no. 4, pp. 633–670

DOI 10.4171/IFB/499