Anisotropic Besov regularity of parabolic PDEs

Abstract

This paper is concerned with the regularity of solutions to parabolic evolution equations. Special attention is paid to the smoothness in the specific anisotropic scale $B_{\tau ,\tau }^{r\mathbf{a}},\frac{1}{\tau }=\frac{r}{d}+\frac{1}{p}$ of Besov spaces where $\mathbf{a}$ measures the anisotropy. The regularity in these spaces determines the approximation order that can be achieved by fully space-time adaptive approximation schemes. In particular, we show that for the heat equation our results significantly improve previous results by Aimar and Gomez [3].