Regularity in $L_p$ Sobolev spaces of solutions to fractional heat equations

TL;DR

This paper establishes sharp regularity results for solutions of fractional heat equations involving nonlocal operators, using symbol calculus and anisotropic function spaces, with applications to stable Lévy processes and boundary conditions.

Contribution

Introduces a symbol calculus for pseudodifferential operators to determine optimal regularity in anisotropic Sobolev and Besov spaces, and analyzes solutions with fractional Laplacians on bounded domains.

Findings

Optimal regularity in anisotropic spaces for solutions on R^n

Unique solutions with specified regularity for fractional Laplacians

Interior regularity improves with smoother forcing functions

Abstract

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations (*) $Pu+\partial_tu=f$ on $\Omega\times I $, where $P$ is a nonlocal operator, and $\Omega \subset R^n$, $I\subset R$. 1) For $P$ a strongly elliptic pseudodifferential operator ($\psi $do) on $R^n$ of order $d\in R_+$, a symbol calculus on $R^{n+1}$ is introduced, that allows showing optimal regularity of solutions in the scale of anisotropic Bessel-potential spaces $H^{(s,s/d)}$, globally over $R^n\times R$, and locally over $\Omega\times I$, for $s\in R$, $1<p<\infty $. Similar results hold in anisotropic Besov spaces $B^{(s,s/d)}$. 2) Let $\Omega $ be smooth bounded, and let $P$ equal $(-\Delta)^a$ ($0<a<1$), or its generalizations to singular integral operators with regular kernels, that are infinitesimal generators of stable L\'evy processes. With the Dirichlet condition $u=0$ on $R^n\setminus\Omega$, the initial condition $u|_{t=0}=0$, and $f\in L_p(\Omega \times I)$, (*) has a unique solution $u\in L_p(I, H_p^{a(2a)}(\bar\Omega))$ with $\partial_tu\in L_p(\Omega \times I)$. Here $H_p^{a(2a)}(\bar\Omega)$ equals $\dot H_p^{2a}(\bar\Omega)$ if $a<1/p$, and is contained in $\dot H_p^{2a-\varepsilon}(\bar\Omega)$ if $a=1/p$, but contains nontrivial elements from $ d^a \bar H_p^{a}(\Omega)$ if $a>1/p$ (where $d(x)= \operatorname{dist}(x,\partial\Omega)$). The interior regularity of $u$ is lifted when $f$ is more smooth.