Higher regularity for parabolic equations based on maximal L_p-L_q spaces

TL;DR

This paper establishes higher regularity results for high-order parabolic equations with general boundary conditions, using maximal L_p-L_q regularity in Besov and Triebel-Lizorkin spaces, emphasizing compatibility conditions for initial data.

Contribution

It introduces a new framework for higher regularity of parabolic equations based on maximal L_p-L_q spaces with Besov and Triebel-Lizorkin spaces, including compatibility conditions.

Findings

Proves isomorphism between solution and data spaces.

Achieves unique smooth solutions under smooth data and compatibility conditions.

Extends regularity theory to general boundary conditions for high-order parabolic equations.

Abstract

In this paper we prove higher regularity for 2m-th order parabolic equations with general boundary conditions. This is a kind of maximal L_p-L_q regularity with differentiability, i.e. the main theorem is isomorphism between the solution space and the data space using Besov and Triebel--Lizorkin spaces. The key is compatibility conditions for the initial data. We are able to get a unique smooth solution if the data satisfying compatibility conditions are smooth.