New maximal regularity results for the heat equation in exterior domains, and applications

TL;DR

This paper establishes new maximal regularity results for the heat equation in various domains using Besov spaces, enabling better control over solutions and applications to nonlinear problems.

Contribution

It introduces novel maximal regularity estimates involving Besov spaces for the heat equation in exterior domains, including mixed norms for low-frequency control.

Findings

Results are similar to whole space or half-space cases for bounded domains.

Use of mixed Besov norms for exterior domains to control low frequencies.

Estimates facilitate global-in-time analysis of nonlinear heat equations.

Abstract

This paper is dedicated to the proof of new maximal regularity results involving Besov spaces for the heat equation in the half-space or in bounded or exterior domains of R^n. We strive for time independent a priori estimates in regularity spaces of type L^1(0,T;X) where X stands for some homogeneous Besov space. In the case of bounded domains, the results that we get are similar to those of the whole space or of the half-space. For exterior domains, we need to use mixed Besov norms in order to get a control on the low frequencies. Those estimates are crucial for proving global-in-time results for nonlinear heat equations in a critical functional framework.