Quasilinear wave equations and microlocal analysis

TL;DR

This paper reviews recent advances in establishing local well-posedness for quasilinear wave equations with low regularity initial data, highlighting the role of microlocal analysis and Strichartz estimates.

Contribution

It introduces new techniques using microlocal analysis to prove well-posedness for wave equations with less regular coefficients than classical methods allow.

Findings

Strichartz estimates for wave operators with Lipschitz coefficients

Bilinear estimates for solutions with low regularity coefficients

Emphasis on microlocal analysis in proving low-regularity well-posedness

Abstract

In this text, we shall give an outline of some recent results (see \ccite{bahourichemin2} \ccite{bahourichemin3} and \ccite{bahourichemin4}) of local wellposedness for two types of quasilinear wave equations for initial data less regular than what is required by the energy method. To go below the regularity prescribed by the classical theory of strictly hyperbolic equations, we have to use the particular properties of the wave equation. The result concerning the first kind of equations must be understood as a Strichartz estimate for wave operators whose coefficients are only Lipschitz while the result concerning the second type of equations is reduced to the proof of a bilinear estimate for the product of two solutions for wave operators whose coefficients are not very regular. The purpose of this talk is to emphasise the importance of ideas coming from microlocal analysis to prove such results.