On local energy decay for large solutions of the Zakharov-Kuznetsov equation

TL;DR

This paper proves local energy decay for large solutions of the Zakharov-Kuznetsov equation in 2D and 3D, extending decay results known for KdV equations and providing decay rates.

Contribution

It establishes local energy decay for global solutions of ZK in both L^2 and H^1 spaces, including decay rates and regions growing unbounded over time.

Findings

Proves local energy decay in suitable regions for ZK solutions.

Extends decay properties from KdV and quartic KdV equations.

Provides decay rates and strong decay results.

Abstract

We consider the Zakharov-Kutznesov (ZK) equation posed in $\mathbb R^d$, with $d=2$ and $3$. Both equations are globally well-posed in $L^2(\mathbb R^d)$. In this paper, we prove local energy decay of global solutions: if $u(t)$ is a solution to ZK with data in $L^2(\mathbb R^d)$, then \[ \liminf_{t\rightarrow \infty}\int_{\Omega_d(t)}u^{2}({\bf x},t)\mathrm{d}{\bf x}=0, \] for suitable regions of space $\Omega_d(t)\subseteq \mathbb R^d$ around the origin, growing unbounded in time, not containing the soliton region. We also prove local decay for $H^1(\mathbb R^d)$ solutions. As a byproduct, our results extend decay properties for KdV and quartic KdV equations proved by Gustavo Ponce and the second author. Sequential rates of decay and other strong decay results are also provided as well.