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arXiv · 2407.12222

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

Sebastian Herr·Robert Schippa·Nikolay Tzvetkov
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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.

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Sebastian Herr, Robert Schippa, Nikolay Tzvetkov. 2024-07-17. The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$. https://arxiv.org/abs/2407.12222

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