Local well-posedness for the Zakharov-Kuznetsov equation on the background of a bounded function

TL;DR

This paper establishes local and global well-posedness results for the 2D Zakharov-Kuznetsov equation on a bounded background, enabling analysis of perturbations around periodic and kink solutions.

Contribution

It extends well-posedness theory for the ZK equation to include backgrounds in $L^ fty$ and proves global results in the energy space, broadening the understanding of solution stability.

Findings

Local well-posedness in $H^s$ for $s\in [1,2]$ on bounded backgrounds

Global well-posedness in $H^1$ energy space

Framework for analyzing perturbations around periodic solutions

Abstract

We prove the local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^s(\mathbb{R}^2)$, for $s\in [1,2]$, on the background of an $L^\infty(\mathbb{R}^3)$-function $\Psi(t,x,y)$, with $\Psi(t,x,y)$ satisfying some natural extra conditions. This result not only gives us a framework to solve the ZK equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of periodic solutions. Additionally, we show the global well-posedness in the energy space $H^1(\mathbb{R}^2)$.