$L^{2}$-estimates for the linear elastic waves

TL;DR

This paper derives optimal $L^{2}$ estimates for elastic wave solutions over large times, establishing bounds that depend minimally on initial data regularity, advancing understanding of wave behavior in elastic media.

Contribution

It provides the first optimal $L^{2}$ bounds for elastic wave solutions with minimal regularity assumptions on initial data.

Findings

Established upper and lower $L^{2}$ bounds for elastic wave components

Demonstrated the bounds hold for large time $t$

Used approximation by smooth auxiliary functions in the proof

Abstract

This paper is concerned with the large time behavior of the solution to the Cauchy problem for the elastic wave equations. In particular, optimal $L^{2}$ estimates of the elastic waves are obtained in the sense that the upper and lower bounds of the $L^{2}$ norms of each component of the solution are proved for large $t$, under the minimum assumptions necessary regarding regularity with respect to initial data. The proof is based on the approximation of the solution by a smooth auxiliary function with suitable parameters.