Regularity of Solutions for the Nonlocal Diffusion Equation on Periodic Distributions

TL;DR

This paper investigates the regularity and convergence properties of solutions to nonlocal diffusion equations with periodic distributions, using Fourier multipliers, and explores how solutions behave with different kernels and initial data.

Contribution

It introduces a unified Fourier multiplier approach to analyze regularity of solutions for nonlocal diffusion equations with various kernels and extends results to super-diffusion operators.

Findings

Solutions exhibit specific regularity depending on initial data and source terms.

Solutions converge to classical diffusion solutions as nonlocality diminishes.

Discontinuities in initial data can persist and decay over time.

Abstract

This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal diffusion equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal super-diffusion operators. We present results on the spatial and temporal regularity of solutions in terms of regularity of the initial data or the diffusion source term. Moreover, solutions of the nonlocal diffusion equation are shown to converge to the solution of the classical diffusion equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension. Furthermore, we show that, for the case of integrable kernels, discontinuities in the initial data propagate and persist in the solution of the nonlocal diffusion equation. The magnitude of a jump discontinuity is shown to decay overtime.