Global Well-Posedness and Global Attractor for Two-dimensional Zakharov-Kuznetsov Equation

TL;DR

This paper proves global well-posedness and the existence of a global attractor for the two-dimensional Zakharov-Kuznetsov equation in certain Sobolev spaces, using the $I$-method and energy increment techniques.

Contribution

It establishes the first global well-posedness results and attractor existence for the 2D Zakharov-Kuznetsov equation in specified Sobolev spaces.

Findings

Global well-posedness in $H^s$ for $rac{5}{7}<s<1$

Existence of global attractor in $H^s$ for $rac{10}{11}<s<1$

Application of $I$-method and energy increment techniques

Abstract

The initial value problem for two-dimensional Zakharov-Kuznetsov equation is shown to be globally well-posed in $H^s({\mathbb{R}^2})$ for all $\frac{5}{7}<s<1$ via using $I$-method in the context of atomic spaces. By means of the increment of modified energy, the exsitence of global attractor for weakly damped, forced Zakharov-Kuznetsov equation is also established in $H^s({\mathbb{R}^2})$ for $\frac{10}{11}<s<1$.