Pointwise convergence for the elastic wave equation

TL;DR

This paper proves that solutions to the elastic wave equation converge pointwise to initial data along every line almost everywhere, given initial data with regularity greater than one-half, establishing near-optimal conditions for convergence.

Contribution

It establishes almost everywhere pointwise convergence of elastic wave solutions to initial data under minimal regularity assumptions, advancing understanding of wave behavior in Sobolev spaces.

Findings

Convergence occurs along every line almost everywhere.

Initial regularity greater than one-half is sufficient for convergence.

The result is nearly optimal in terms of regularity requirements.

Abstract

We study pointwise convergence of the solution to the elastic wave equation to the initial data which lies in the Sobolev spaces. We prove that the solution converges along every lines to the initial data almost everywhere whenever the initial regularity is greater than one half. We show this is almost optimal.