Global results for weakly dispersive KP-II equations on the cylinder

TL;DR

The paper addresses global well-posedness for the dispersion-generalized KP-II equation on the cylinder in the weakly dispersive regime $\alpha<2$, focusing on real-valued initial data in $L^2(\mathbb{R}\times\mathbb{T})$. It develops a framework of short-time Fourier restriction spaces built from $\ell^2$-decoupling-based linear Strichartz estimates, combined with bilinear and trilinear convolution bounds to control the nonlinear evolution. A frequency-dependent time localization, resonance analysis, and a frequency-envelope method yield global well-posedness for $\alpha$ in $(\alpha^*,2)$ and a long-time decay property for small mass, with quantitative bounds in anisotropic Sobolev scales. The results extend KP-II theory to partially periodic settings, offering a robust approach that leverages decoupling, short-time analysis, and commutator-based energy methods to handle the quasilinear nature and indefinite energy.

Abstract

We consider the dispersion-generalized KP-II equation on a partially periodic domain in the weakly dispersive regime. We use Fourier decoupling techniques to derive essentially sharp Strichartz estimates. With these at hand, we show global well-posedness of the quasilinear Cauchy problem in $L^2(\mathbb{R} \times \mathbb{T})$. Finally, we prove a long-time decay property of solutions with small mass by using the Kato smoothing effect in the fractional case.

Figures (1)

• Figure 1: Anisotropic scaling transforms $\delta$-flat sets.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Corollary 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Corollary 2.5
• Remark 2.6