The Zakharov-Kuznetsov equation in high dimensions: Small initial data of critical regularity

TL;DR

This paper proves global well-posedness and scattering for the high-dimensional Zakharov-Kuznetsov equation with small initial data in critical spaces, using novel endpoint estimates and bilinear restriction techniques.

Contribution

It introduces new endpoint non-isotropic Strichartz estimates and applies them to establish global results for the Zakharov-Kuznetsov equation in dimensions five and higher.

Findings

Global well-posedness for small data in critical spaces

Solutions scatter to free solutions as time approaches infinity

Extension of results to dimension four under radiality assumption

Abstract

The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces and it is proved that solutions scatter to free solutions as $t \to \pm \infty$. The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schr\"odinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension $d=4$.