Low regularity analysis of the Zakharov--Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$

TL;DR

This paper advances the understanding of the Zakharov-Kuznetsov equation on a cylindrical domain by establishing optimal local well-posedness results at low regularity and introducing innovative methods for analyzing resonant interactions.

Contribution

It improves local well-posedness to regularity s > 1/2, identifies geometric properties of the resonant set, and develops a novel approach for solutions with randomized initial data at negative regularity.

Findings

Optimal local well-posedness for s > 1/2

Solutions for generic data in H^s with s < 0 under randomization

Enhanced understanding of the resonant set's geometric structure

Abstract

We consider the Cauchy problem for the Zakharov-Kuznetsov equation in the cylinder. We improve the local wellposedness to spaces of regularity $s > 1/2$. The result is optimal in terms of the corresponding bilinear estimate or Picard iteration. Our method is based on an improvement of the understanding of the resonant set, identifying and exploiting its particular geometric properties. We also consider the problem under randomization of the initial data, in which case we obtain solutions for generic data in $H^{s}$ for some $s < 0$. To do so, we consider a novel approach based on lower regularity modifications of the classical $X^{s, b}$ spaces that allow to control concentration of mass in small sets of frequencies.