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

This paper considers the problem of optimal well-posedness for general quasi-linear wave equations in R1+3 of the type (1.1). In general, equations of this type are ill-posed with Hs data for s ≤ 2. The optimal result of the well-posedness with data in Hs, s > 2 was proved by Smith–Tataru by constructing parametrices using wave packets. In this paper we give the proof by the vectorfield approach. This approach is initiated by Klainerman, developed by him and Rodnianski to achieve the result of s > 2 + 2−√3/2 and then applied to the Einstein vacuum equations to achieve the result of s > 2. To achieve the optimal result for the general quasi-linear wave equations, one has to face the major hurdle caused by the Ricci tensor of the metric. This posed a question that if the geometric approach can provide the sharp result for the non-geometric equations. The new ingredients of this paper concern the regularity properties of the eikonal equation associated to (1.1). The optimal result for (1.1) is achieved based on geometric normalization and new observations on the Raychaudhuri equation and the mass aspect function