Minimal regularity solutions of nonlinear wave equations

Abstract

Inspired by the need to understand the complex systems of non-linear wave equations which arise in physics, there has recently been much interest in proving existence and uniqueness for solutions of nonlinear wave equations with low regularity initial data. We give counterexamples to local existence with low regularity data for the typical nonlinear wave equations. In the semi-linear case these are sharp, in the sense that with slightly more regularity one can prove local existence. We also present join work with Georgiev and Sogge proving global existence for a certain class of semi-linear wave equation. This result was a conjecture of Strauss following an initial result of Fritz John. We develop weighted Strichartz estimates whose proof uses techniques from harmonic analysis taking into account the symmetries of the wave equation.