A regularity theory for evolution equations with space-time anisotropic non-local operators in mixed-norm Sobolev spaces

TL;DR

This paper develops a regularity theory for solutions to complex space-time non-local evolution equations with anisotropic operators, using probabilistic methods and interpolation theory to handle singularities and initial data characterization.

Contribution

It introduces a novel probabilistic approach to analyze regularity of anisotropic non-local evolution equations in mixed-norm Sobolev spaces, addressing Fourier singularities and initial data spaces.

Findings

Established existence and uniqueness of solutions.

Derived precise regularity estimates for solutions.

Identified optimal initial data spaces using interpolation.

Abstract

In this article, we study the regularity of solutions to inhomogeneous time-fractional evolution equations involving anisotropic non-local operators in mixed-norm Sobolev spaces of variable order, with non-trivial initial conditions. The primary focus is on space-time non-local equations where the spatial operator is the infinitesimal generator of a vector of independent subordinate Brownian motions, making it the sum of subdimensional non-local operators. A representative example of such an operator is $(\Delta_{x})^{\beta_{1}/2}+(\Delta_{y})^{\beta_{2}/2}$. We establish existence, uniqueness, and precise estimates for solutions in corresponding Sobolev spaces. Due to singularities arising in the Fourier transforms of our operators, traditional methods involving Fourier analysis are not directly applicable. Instead, we employ a probabilistic approach to derive solution estimates. Additionally, we identify the optimal initial data space using generalized real interpolation theory.