Ill-posedness of a quasilinear wave equation in two dimensions for data in $H^{7/4}$

TL;DR

This paper demonstrates the sharp threshold for well-posedness of a specific quasilinear wave equation in two dimensions, showing ill-posedness at a critical regularity level, refining previous results.

Contribution

It establishes a sharp ill-posedness result for a quasilinear wave equation at the critical Sobolev regularity, extending and refining earlier well-posedness findings.

Findings

Ill-posedness at critical regularity $H^{11/4}$

Construction of initial data causing ill-posedness

Refinement of previous well-posedness thresholds

Abstract

In this article, we study the ill-posedness of a quasilinear wave equation. It was shown by Tataru and Smith in 2005 that for any $s>7/4$ (or $11/4$ in our situation), the equation is well-posed in $H^{s}\times H^{s-1}$. We show a sharpness result by exhibiting a quasilinear wave equation and an initial data such that the Cauchy problem is ill-posed for in $H^{11/4} (\ln H)^{-\beta}\times H^{7/4} (\ln H^{-\beta})$.