Decay estimates of wave equations in un-isotropic media

TL;DR

This paper establishes decay estimates for solutions to non-isotropic wave equations, leveraging commutation properties, and applies these results to prove global regularity for small data in certain nonlinear systems, with potential applications in crystal optics.

Contribution

It introduces decay estimates based on commutation with the scaling vector field and provides simplified proofs for global regularity in non-isotropic wave systems.

Findings

Decay estimates depend only on commutation with the scaling vector field.

Simplified proofs for small data global regularity in non-isotropic wave systems.

Potential relevance to biaxial refraction in crystal optics.

Abstract

We prove decay estimates for solutions to non-isotropic linear systems of wave equations. The defining feature of these estimates is that they depend only on the commutation properties of the system with the scaling vector field. As application we give two surprisingly simple proofs for small data global regularity results non-isotropic systems of wave equations in $\mathbb{R}^{1+3}$ with cubic semilinear nonlinearities. We hope that the techniques presented here are relevant for the more difficult and important case of biaxial refraction in crystal optics.