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arXiv · 2208.01306

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

Tatsu-Hiko Miura
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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.

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Tatsu-Hiko Miura. 2022-08-02. Error estimate for classical solutions to the heat equation in a moving thin domain and its limit equation. https://arxiv.org/abs/2208.01306

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