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(Φ)∂\a\b2Φ=N(Φ,∂Φ) are locally well-posed in H2+ϵ×H1+ϵ(R3). The sharpness of this result was immediately known, given an older result due to Lindblad in \cite{QuasiIllP} which showed that the equation \BBoxmΦ=−Φ(\Lb(Flat)2Φ) is illposed in H2×H1(R3). 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 H2×H1(R3). Indeed, as with shock formation, this is actually a generic phenomenon: for almost all (g−1)\a\b(Φ) which are perturbations of the Minkowski metric we can specify initial data which is arbitrarily small in H2×H1(R3) but whose 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 H3×H2(R3) is an exceptional case: every other equation in its class of irrotational, compressible relativistic fluid equations is illposed in H3×H2(R3).