Anisotropic Besov regularity of parabolic PDEs

TL;DR

This paper investigates the anisotropic Besov regularity of solutions to parabolic PDEs, revealing how this regularity influences approximation schemes and improving upon previous results for the heat equation.

Contribution

It introduces a detailed analysis of anisotropic Besov regularity for parabolic PDEs, enhancing understanding of solution smoothness and approximation capabilities.

Findings

Regularity in anisotropic Besov spaces determines approximation order.

Results significantly improve previous regularity estimates for the heat equation.

Provides a framework for analyzing anisotropic smoothness in parabolic PDE solutions.

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^{r\mathbf{a}}_{\tau,\tau}, \ \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].