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Ill-posedness of a quasilinear wave equation in two dimensions for data in $H^{7/4}$ (2107.03732v2)
Published 8 Jul 2021 in math.AP
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 $H{11/4} (\ln H){-\beta}\times H{7/4} (\ln H{-\beta})$.