Local well-posedness for the quasilinear Schr\"odinger equations via the generalized energy method

TL;DR

This paper introduces a generalized energy method combining momentum and energy estimates to establish local well-posedness for quasilinear Schrödinger equations, extending results to large data and low regularity cases.

Contribution

It develops a new generalized energy method that unifies viscosity and dispersive techniques for quasilinear Schrödinger equations, improving existing well-posedness results.

Findings

Established local well-posedness for quadratic interactions at low regularity.

Extended low regularity results to cubic interactions with small initial data.

Unified treatment for small and large data cases using the generalized energy method.

Abstract

We study the Cauchy problem of quasilinear Schr\"odinger equations, for which Kenig et al. (Invent Math, 2004; Adv Math, 2006) obtained large data local well-posedness by pseudo-differential techniques and viscosity methods, while Marzuola et al. (Adv Math, 2012; Kyoto J Math, 2014; Arch Ration Mech Anal, 2021) and Ben et al. (Arch Ration Mech Anal, 2024) improved the results by dispersive arguments. In this paper, we introduce a generalized energy method that combines momentum and energy estimates to close the bounds, thereby obtaining our results through viscosity methods. If the data is small, the proof relies mainly on integration by parts and Sobolev embeddings, much like the classical local existence theory for semilinear Schr\"odinger equations. For large data, the framework remains applicable with the incorporation of certain pseudo-differential tools. In the case of quadratic interactions, we establish low regularity local well-posedness for both small and large data in the same function spaces as in works of Kenig et al. For cubic interactions with small initial data, we recover the low regularity results obtained by Marzuola et al. (Kyoto J Math, 2014).