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

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 $H^{11/4} (\ln H)^{-\beta}\times H^{7/4} (\ln H^{-\beta})$.