Fractional Diffusion in the full space: decay and regularity

TL;DR

This paper studies fractional PDEs on the entire space, establishing existence, uniqueness, decay rates, and regularity of solutions, which supports the development of numerical methods like FEM-BEM coupling.

Contribution

It introduces a rigorous framework for fractional PDEs on full space, including convergence of truncated problems and decay estimates, advancing numerical analysis techniques.

Findings

Solutions exist and are unique for the fractional PDEs considered.

Solutions to truncated problems converge to the full space solution as the truncation parameter grows.

The paper provides decay rates and regularity estimates for solutions.

Abstract

We consider fractional partial differential equations posed on the full space $\R^d$. Using the well-known Caffarelli-Silvestre extension to $\R^d \times \R^+$ as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on $\R^d \times (0,\YY)$ converge to the solution of the original problem as $\YY \rightarrow \infty$. Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of numerical methods for the full space problem, such as FEM-BEM coupling techniques.