Quasilinear Schr\"odinger equations I: Small data and quadratic interactions

TL;DR

This paper establishes local well-posedness for general quasilinear Schrödinger equations in low-regularity Sobolev spaces using dispersive methods, improving upon previous high-regularity results.

Contribution

It introduces dispersive techniques to prove well-posedness in low-regularity spaces, advancing the understanding of quasilinear Schrödinger equations beyond viscosity-based methods.

Findings

Proved local well-posedness in low-regularity Sobolev spaces.

Developed new function spaces inspired by prior dispersive analysis.

Extended previous results by reducing regularity requirements.

Abstract

In this article we prove local well-posedness in low-regularity Sobolev spaces for general quasilinear Schr\"odinger equations. These results represent improvements of the pioneering works by Kenig-Ponce-Vega and Kenig-Ponce-Rolvung-Vega, where viscosity methods were used to prove existence of solutions in very high regularity spaces. Our arguments here are purely dispersive. The function spaces in which we show existence are constructed in ways motivated by the results of Mizohata, Ichinose, Doi, and others, including the authors.