Regularity of solutions to space--time fractional wave equations: a PDE approach

TL;DR

This paper studies the regularity and well-posedness of space-time fractional wave equations involving fractional elliptic operators and Caputo derivatives, providing new regularity estimates that impact numerical analysis approaches.

Contribution

It offers new existence, uniqueness, and regularity results for space-time fractional wave equations, highlighting issues with standard assumptions in numerical methods.

Findings

Established existence and uniqueness of solutions.

Derived regularity estimates in time and space.

Identified limitations of common numerical assumptions.

Abstract

We consider an evolution equation involving the fractional powers, of order $s \in (0,1)$, of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order $\gamma \in (1,2]$. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi--stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi--infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time--regularity results show that the usual assumptions made in the numerical analysis literature are problematic