Wellposedness of some quasi-linear Schrödinger equations

Abstract

This article is devoted to the study of a quasilinear Schrodinger equation coupled with an elliptic equation on the metric g. We first prove that, in this context, the propagation of regularity holds which ensures local wellposedness for initial data small enough in \(\dot H^{\tfrac{1} {2}} \) and belonging to the Besov space \(\dot B_{2,1}^{\tfrac{3} {2}} \). In a second step, we establish Strichartz estimates for time dependent rough metrics to obtain a lower bound of the time existence which only involves the \(\dot B_{2,\infty }^{1 + \varepsilon } \) norm on the initial data.