A geometric approach for sharp Local well-posedness of quasilinear wave equations

Abstract

The commuting vector fields approach, devised for strichartz estimates in [13], was developed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\mathbb{R}}^{1+3}$ by Klainerman and Rodnianski. Via this approach they obtained the local well-posedness in $H^s$ with $s>2$ for $(1+3)$ vacuum Einstein equations, by taking advantage of the vanishing Ricci curvature. The sharp, $H^{2+\epsilon }$, local well-posedness result for general quasilinear wave equation was achieved by Smith and Tataru by constructing a parametrix using wave packets. Using the vector fields approach, one has to face the major hurdle caused by the Ricci tensor of the metric for the quasi-linear wave equations. This posed a question that if the geometric approach can provide the sharp result for the non-geometric equations. In this paper, based on geometric normalization and new observations on the mass aspect function, we prove the sharp local well-posedness of general quasilinear wave equation in ${\mathbb{R}}^{1+3}$ by a vector field approach.

References in corpus (2)

• Parametrix for wave equations on a rough background I: regularity of the phase at initial time
• On Ricci coefficients of null hypersurfaces with time foliation in Einstein vacuum space-time

Cited by in corpus (1)

• An intrinsic hyperboloid approach for Einstein Klein-Gordon equations