Stability results for nonlinear hyperbolic equations

Abstract

This thesis is concerned with three stability results concerning nonlinear wave equations described over four Chapters, the first one being introductory. A common theme among all three results is that we must extend the vector field method to situations where we do not have direct access to the usual collection of operators for one reason or another. Of the three results, we provide summaries for the first two and complete proofs for the third.

The second Chapter in the thesis describes joint work with Federico Pasqualotto concerning global stability of the trivial solution to quasilinear wave equations satisfying the null condition under a novel class of perturbations. These perturbations are allowed to be localized around a finite collection of points, in contrast with the classical theory, which requires data to be localized around a single point. The proof uses an analysis of the interaction of waves originating from distant points and adapting the vector.

The third Chapter in this thesis concerns joint work with Samuel Zbarsky concerning the stability and instability of plane wave solutions to semilinear systems of wave equations satisfying the null condition. We identify sets of conditions leading to both stability and instability. The stability proof requires analyzing the geometry of the interaction between the background plane wave and the perturbation while the proof of the instability result is based on a geometric optics ansatz.

The fourth and most substantial Chapter of this thesis initiates the study of stability results for anisotropic systems of quasilinear wave equations. Equations of this kind arise naturally in the study of birefringent phenomena, and in particular in the study of crystal optics. The proof introduces a strategy for proving decay based on bilinear energy estimates that we interface with the vector field method and several geometric considerations. The study of this problem represents the first step in a research program envisioned by Klainerman involving the study of hyperbolic equations having multiple characteristics. In addition to providing proofs of the main results, we discuss the role of anisotropic systems of wave equations in this larger context.