Regularity of the solution to fractional diffusion, advection, 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.

40 pages

References in corpus (4)

• Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics
• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces
• Gevrey regularity for integro-differential operators

Cited by in corpus (1)

• Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions