Global stability for a nonlinear system of anisotropic wave equations

TL;DR

This paper establishes the global stability of anisotropic quasilinear wave systems, which model phenomena like crystal optics, by developing a physical space approach combining energy estimates and vector field methods.

Contribution

It introduces a novel physical space strategy for proving decay in anisotropic wave systems, integrating bilinear energy estimates with the vector field method.

Findings

Proves decay for nonlinear anisotropic wave equations.

Develops a new approach combining energy estimates and vector fields.

Analyzes spacetime geometry interactions of waves.

Abstract

In this paper, we initiate the study of global stability for anisotropic systems of quasilinear wave equations. Equations of this kind arise naturally in the study of crystal optics, and they exhibit birefringence. We introduce a physical space strategy based on bilinear energy estimates that allows us to prove decay for the nonlinear problem. This uses decay for the homogeneous wave equation as a black box. The proof also requires us to interface this strategy with the vector field method and take advantage of the scaling vector field. A careful analysis of the spacetime geometry of the interaction between waves is necessary in the proof.