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

Mykael Cardoso, Gleison do N. Santos, Roger P. de Moura

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.

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

Abstract

In this paper we establish exponential decay results for solutions of the damped -dimensional Zakharov--Kuznetsov equation for . More precisely, we prove the exponential decay of the norm when the damping is localized. In addition, when the dissipative mechanism acts on the whole space , we prove the exponential decay of the norm. Our strategy of proof combines a Kato's type smoothing effect, unique continuation and an observability inequality.
Paper Structure (9 sections, 12 theorems, 108 equations)

This paper contains 9 sections, 12 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Notation and Auxiliary Results
  3. Notation
  4. Technical Lemmas
  5. Well-posedness and the observability inequality
  6. Proof of the main results
  7. Proof of Theorem \ref{['teo1']}
  8. Proof of Theorem \ref{['teo2']}
  9. Appendix

Key Result

Theorem 1.1

Let $a \in W^{1,\infty}(\mathbb{R}^n)$ be a non-negative function satisfying for some $R, \alpha_0 > 0$. Then, given $L > 0$, there exist positive constants $\delta = \delta(L)$ and $C = C(L)$ such that for any mild solution $u \in X_T^1$ of eqZK1dampped with $u_0 \in H^1(\mathbb{R}^n)$ satisfying $\|u_0\|_{L^2(\mathbb{R}^n)} \leq L$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem A
  • Theorem 1.2
  • Lemma 2.1: Gagliardo--Nirenberg inequality
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • ...and 9 more