Strichartz estimates and local regularity for the elastic wave equation with singular potentials

TL;DR

This paper establishes weighted $L^2$ and Strichartz estimates for the elastic wave equation with singular potentials, improving previous results by relaxing conditions and demonstrating local regularity of solutions.

Contribution

It introduces new weighted estimates for the elastic wave equation with singular potentials and derives Strichartz estimates under weaker ellipticity conditions.

Findings

Weighted $L^2$ estimates for elastic wave equations with singular potentials

Strichartz estimates under minimal ellipticity assumptions

Local regularity of solutions to the elastic wave equation

Abstract

We obtain weighted $L^2$ estimates for the elastic wave equation perturbed by singular potentials including the inverse-square potential. We then deduce the Strichartz estimates under the sole ellipticity condition for the Lam\'e operator $-\Delta^\ast$. This improves upon the previous result in \cite{BFRVV} which relies on a stronger condition to guarantee the self-adjointness of $-\Delta^\ast$. Furthermore, by establishing local energy estimates for the elastic wave equation we also prove that the solution has local regularity.