Regularity of the solution to fractional diffusion, advection, reaction equations

TL;DR

This paper studies the regularity properties of solutions to fractional diffusion, advection, and reaction equations on bounded domains, revealing how solution smoothness depends on parameters and boundary behavior.

Contribution

The paper introduces new regularity results for fractional PDEs in weighted Sobolev spaces, highlighting the impact of parameters and boundary conditions on solution smoothness.

Findings

Solution regularity is bounded by endpoint behavior influenced by parameters.

Regularity of fractional diffusion reaction solutions is lower than pure diffusion.

Solutions to advection-reaction equations are two orders less regular than those to diffusion-reaction equations.

Abstract

In this report we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in $\mathbb{R}^{1}$. The analysis is performed in the weighted Sobolev spaces, $H_{(a , b)}^{s}(\mathrm{I})$. Three different characterizations of $H_{(a , b)}^{s}(\mathrm{I})$ are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded by the endpoint behavior of the solution, which is determined by the parameters $\alpha$ and $r$ defining the fractional diffusion operator. Additionally, the analysis shows that for a sufficiently smooth right hand side function, the regularity of the solution to fractional diffusion reaction equation is lower than that of the fractional diffusion equation. Also, the regularity of the solution to fractional diffusion advection reaction equation is two orders lower than that of the fractional diffusion reaction equation.