The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$

TL;DR

This paper extends well-posedness results for the KP-II equation on a torus to initial data with negative Sobolev regularity by developing new linear Strichartz estimates that leverage recent decoupling inequalities.

Contribution

It introduces a novel linear $L^4$-Strichartz estimate for the KP-II equation on $ ext{T}^2$, combining decoupling inequalities with semiclassical estimates to handle rough initial data.

Findings

Established $L^2$-wellposedness for initial data below $L^2$ regularity.

Developed a new linear $L^4$-Strichartz estimate using decoupling and semiclassical techniques.

Extended Bourgain's bilinear estimate to frequency-dependent times.

Abstract

We extend Bourgain's $L^2$-wellposedness result for the KP-II equation on $\mathbb{T}^2$ to initial data with negative Sobolev regularity. The key ingredient is a new linear $L^4$-Strichartz estimate which is effective on frequency-dependent time scales. The $L^4$-Strichartz estimates follow from combining an $\ell^2$-decoupling inequality recently proved by Guth--Maldague--Oh with semiclassical Strichartz estimates. Moreover, we rely on a variant of Bourgain's bilinear Strichartz estimate on frequency-dependent times, which is proved via the C\'ordoba--Fefferman square function estimate.