Sharp local well-posedness for quasilinear wave equations with spherical symmetry

Abstract

In this paper, we prove a sharp local well-posedness result for spherically symmetric solutions to quasilinear wave equations with rough initial data, when the spatial dimension is three or higher. Our approach is based on Morawetz type local energy estimates with fractional regularity for linear wave equations with variable $C^1$ coefficients, which rely on the multiplier method, weighted Littlewood–Paley theory, duality and interpolation. Together with weighted linear and nonlinear estimates (including weighted trace estimates, Hardy’s inequality, a fractional chain rule and a fractional Leibniz rule) which are adapted to the problem, the well-posedness result is proved by iteration. In addition, our argument yields almost global existence for $n=3$ and global existence for $n\ge 4$ when the initial data are small and spherically symmetric with almost critical Sobolev regularity.