Quasilinear Schr\"odinger equations II: Small data and cubic nonlinearities

TL;DR

This paper extends the analysis of quasilinear Schrödinger equations to cubic nonlinearities, demonstrating small data well-posedness in Sobolev spaces without the need for summability conditions used in quadratic cases.

Contribution

It shows that for cubic interactions, the well-posedness results improve by removing the $l^1$ summability condition, leveraging the inherent $L^2$ structure of Schrödinger equations.

Findings

Established small data well-posedness in $H^s$ spaces for cubic nonlinearities.

Removed the necessity of $l^1$ summability over cubes in the cubic case.

Extended previous quadratic interaction results to cubic interactions with improved conditions.

Abstract

In part I of this project we examined low regularity local well-posedness for generic quasilinear Schr\"odinger equations with small data. This improved, in the small data regime, the preceding results of Kenig, Ponce, and Vega as well as Kenig, Ponce, Rolvung, and Vega. In the setting of quadratic interactions, the (translation invariant) function spaces which were utilized incorporated an $l^1$ summability over cubes in order to account for Mizohata's integrability condition, which is a necessary condition for the $L^2$ well-posedness for the linearized equation. For cubic interactions, this integrability condition meshes better with the inherent $L^2$ nature of the Schr\"odinger equation, and such summability is not required. Thus we are able to prove small data well-posedness in $H^s$ spaces.