On the Local Well Posedness of Quasilinear Schrödinger Equations in Arbitrary Space Dimension

Abstract

A class of quasilinear Schrödinger equations is studied which contain strongly singular nonlinearities including derivatives of second order. Such equations have been derived as models of several physical phenomena. The results of this paper apply to the superfluid film equation in fluid mechanics. The local well posedness for smooth solutions of the Cauchy problem is proved in arbitrary space dimension without any smallness assumption on the initial value. This improves results in the literature which cover the one-dimensional case only. Almost global well posedness results are obtained as well. The proof combines Nash–Moser techniques on implicit function theorems in Fréchet spaces with hyperbolic semigroup theory including evolution systems using the intersection of all Sobolev spaces as basic function space. The nondissipativity is overcome by a transformation procedure yielding the evolution operator and a priori estimates of higher order Sobolev norms for the linearized problem. Here a certain trace has to vanish in view of apparent integrability conditions. The transformations and the regularity theory are substantially different from the easier one-dimensional case. In particular, continuity estimates are proved for the operator norms of the linear evolution operator in negative Sobolev spaces.