On the stabilization of $L^2$ and $H^1$ norms for the Zakharov-Kuznetsov equation with damping

TL;DR

This paper proves exponential decay of solutions in $L^2$ and $H^1$ norms for the damped Zakharov-Kuznetsov equation in 2D and 3D, using smoothing, unique continuation, and observability techniques.

Contribution

It establishes new exponential decay results for the damped Zakharov-Kuznetsov equation in multiple dimensions with localized and full-space damping.

Findings

Exponential decay of $L^2$ norm with localized damping.

Exponential decay of $H^1$ norm with full-space damping.

Combines smoothing effects, unique continuation, and observability in proofs.

Abstract

In this paper we establish exponential decay results for solutions of the damped $n$-dimensional Zakharov--Kuznetsov equation for $2 \le n \le 3$. More precisely, we prove the exponential decay of the $L^2(\mathbb{R}^n)$ norm when the damping is localized. In addition, when the dissipative mechanism acts on the whole space $\mathbb{R}^n$, we prove the exponential decay of the $H^1(\mathbb{R}^n)$ norm. Our strategy of proof combines a Kato's type smoothing effect, unique continuation and an observability inequality.