Quasilinear Schr\"odinger Equations

TL;DR

This paper establishes local well-posedness for quasilinear Schrödinger equations with initial data in unweighted Sobolev spaces, removing smallness constraints through novel smoothing estimates.

Contribution

It introduces an uncentered Doi's Lemma and extends previous work to prove well-posedness without small data assumptions.

Findings

Proved local well-posedness for quasilinear Schrödinger equations.

Developed a uncentered Doi's Lemma for smoothing estimates.

Removed smallness condition for initial data in unweighted Sobolev spaces.

Abstract

In this paper we prove local well-posedness for Quasi-linear Scrh\"odinger 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. 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 to remove the smallness condition in unweighted spaces. This is accomplished by developing a uncentered 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.