Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher

TL;DR

This paper establishes local well-posedness for the Zakharov-Kuznetsov equation in dimensions three and higher within the subcritical Sobolev space range, leading to global results in specific cases.

Contribution

It proves the optimal local well-posedness range for the Zakharov-Kuznetsov equation in higher dimensions, extending previous results.

Findings

Local well-posedness in $H^s$ for $s>(d-4)/2$

Global well-posedness in $L^2$ for $d=3$

Global well-posedness in $H^1$ for $d=4$ under smallness condition

Abstract

The Zakharov-Kuznetsov equation in space dimension $d\geq 3$ is considered. It is proved that the Cauchy problem is locally well-posed in $H^s(\mathbb{R}^d)$ in the full subcritical range $s>(d-4)/2$, which is optimal up to the endpoint. As a corollary, global well-posedness in $L^2(\mathbb{R}^3)$ and, under a smallness condition, in $H^1(\mathbb{R}^4)$, follow.