Well-Posedness for the Two-dimensional Zakharov-Kuznetsov Equation

TL;DR

This paper proves the global well-posedness of the 2D Zakharov-Kuznetsov equation in certain Sobolev spaces using the I-method, and also establishes local well-posedness for a symmetrized version in atomic spaces.

Contribution

It introduces new well-posedness results for the 2D Zakharov-Kuznetsov equation in specific function spaces, extending previous understanding of the equation's behavior.

Findings

Global well-posedness in $H^{s}$ for $rac{11}{13}<s<1$

Local well-posedness in $B^{1/2}_{2,1}$ for the symmetrized ZK equation

Application of the I-method and atomic space techniques

Abstract

We show the global well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^{s}({\mathbb{R}^2})$ when $\frac{11}{13}<s<1$ via the I-method. Additionally, local well-posedness for the symmetrized ZK equation in $ B^\frac{1}{2}_{2,1}(\mathbb{R}^2)$ is established by using atomic spaces.