Asymptotically Stable Ill-Posedness of Geometric Quasilinear Wave Equations

Abstract

It has been known since the work of Smith and Tataru in \cite{TataruLWP} that quasilinear wave equations $(g^{-1})^{\a\b}(\Phi)\partial^{2}_{\a\b}\Phi=\mathcal{N}(\Phi,\partial\Phi)$ are locally well-posed in $H^{2+\epsilon}\times H^{1+\epsilon}(\mathbb{R}^{3})$. The sharpness of this result was immediately known, given an older result due to Lindblad in \cite{QuasiIllP} which showed that the equation $\BBox_{m}\Phi=-\Phi(\Lb^{2}_{(Flat)}\Phi)$ is illposed in $H^{2}\times H^{1}(\mathbb{R}^{3})$. We show that the recent work of Speck, Holzegel, Luk and Wong in \cite{ShockPlane} on nearly plane symmetric shock formation is intimately connected to the low-regularity illposedness of geometric quasilinear wave equations in $H^{2}\times H^{1}(\mathbb{R}^{3})$. Indeed, as with shock formation, this is actually a generic phenomenon: for almost all $(g^{-1})^{\a\b}(\Phi)$ which are perturbations of the Minkowski metric we can specify initial data which is arbitrarily small in $H^{2}\times H^{1}(\mathbb{R}^{3})$ but whose $\dot{H}^{1}$ energy blows up arbitrarily fast in the domain of future dependence of the data. We demonstrate that illposedness is actually a corollary of the nearly planar shock formation theorem in $3+1$ dimensions. The nearly planar shock formation result actually gives us even more: the stability of the breakdown under asymptotically small perturbations of ``Lindblad-type" initial data. In other words, Lindblad's result is not simply an artifact of symmetry. We then show that the proof in \cite{ShockPlane} extends from $2+1$ to $3+1$ dimensions. This is largely the same argument, although elliptic estimates are now necessary to control some of the new top-order error terms which are nontrivial in the $3+1$ dimensional setting. In order to control these terms we follow the structure presented in \cite{ShockSpeck}. Our result demonstrates that the conjectural local well-posedness of the timelike minimal surface equation in $H^{3}\times H^{2}(\mathbb{R}^{3})$ is an exceptional case: every other equation in its class of irrotational, compressible relativistic fluid equations is illposed in $H^{3}\times H^{2}(\mathbb{R}^{3})$.