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

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

-
Qian Wang , University of Oxford, England
Fine Hall 110

Please note special location and time.   The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed 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 him 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. I will present my recent work, which proves the sharp local well-posedness of general quasilinear wave equation in ${\Bbb R}^{1+3}$ by a vector field approach,  based on geometric normalization and new observations on the mass aspect functions.