Strichartz and uniform Sobolev inequalities for the elastic wave equation

TL;DR

This paper extends Strichartz estimates from classical to elastic wave equations, providing endpoint results and uniform Sobolev inequalities by diagonalizing the Lamé operator, simplifying previous proofs.

Contribution

It introduces a method to derive dispersive and Strichartz estimates for elastic waves, including endpoint cases, and establishes uniform Sobolev inequalities, advancing analysis of elastic wave equations.

Findings

Extended Strichartz estimates to elastic wave equations

Derived uniform Sobolev inequalities for elastic operators

Provided simplified proofs via diagonalization of Lamé operator

Abstract

We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lam\'e operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.