Quasilinear Schrödinger Equations

Abstract

In this paper we prove local well-posedness for Quasi-linear Scrhödinger equations with initial data in unweighted Sobolev Spaces. For small initial data with minimal smoothness this has addressed by J. Marzuola, J. Metcalfe and D. Tataru [15], [16]. This work does not attempt to address the minimal regularity for initial data, but instead builds on the previous results of C. Kenig, G. Ponce, and L. Vega [13], [12] to remove the smallness condition in unweighted spaces. This is accomplished by developing a non-centered version of Doi's Lemma, which allows one to prove Kato type smoothing estimates. These estimates make it possible to achieve the necessary a priori linear results.