Besov regularity of non-linear parabolic PDEs on non-convex polyhedral domains

TL;DR

This paper investigates the Besov space regularity of solutions to semilinear parabolic PDEs on non-convex domains, demonstrating sufficient smoothness to support adaptive approximation methods.

Contribution

It establishes high Besov regularity for solutions of non-linear parabolic PDEs on complex domains, justifying the effectiveness of adaptive algorithms.

Findings

Besov regularity is sufficient for adaptive schemes.

Regularity results apply to non-convex polyhedral domains.

Proofs use Schauder's fixed point theorem.

Abstract

This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non-convex domains. Special attention is paid to the smoothness in the specific scale $B^r_{\tau,\tau}$, $\frac{1}{\tau}=\frac rd+ \frac 1p$ of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our proofs are based on Schauder's fixed point theorem.