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

TL;DR

This work advances the local well-posedness theory for the periodic KP-II equation on $\mathbb{T}^2$ into negative Sobolev spaces by marrying short-time Fourier restriction analysis with sharp linear and nonlinear estimates. The centerpiece is a novel $L^4$ Strichartz bound derived from $\ell^2$-decoupling, complemented by frequency-dependent time localization and bilinear/trilinear estimates that control energy and differences of solutions at low regularity. The main result establishes local well-posedness for $s > -\tfrac{1}{90}$, with a Lipschitz-continuous data-to-solution map in a refined topology, and without relying on complete integrability. The methods also accommodate irrational tori and cylindrical geometries, highlighting the versatility of decoupling-inspired short-time analysis for dispersive PDEs in periodic settings.

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 Córdoba--Fefferman square function estimate.

Figures (1)

• Figure 1: Galilean transformation to shift the long direction into the $\xi$-direction.

Theorems & Definitions (54)

• Theorem 1.1: Local well-posedness below $L^2$
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5: IonescuKenigTataru2008
• Lemma 2.6: Linear energy estimate
• Lemma 2.7: GuoOh2018
• Definition 3.1: $\delta$-flat set
• Theorem 3.2: GuthMaldagueOh2024