Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$

Zhaohi  Huo; Yueling Jia; Qiaoxin  Li

doi:10.3934/dcdss.2016075

Article Contents

2016, Volume 9, Issue 6: 1797-1851. Doi: 10.3934/dcdss.2016075

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Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$

1.
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190
2.
Institute of Applied Physics and Computational Mathematics, Beijing, 100094
3.
China Academy of Aerospace Aerodynamics, Beijing 100074

Received: July 2015

Revised: September 2016

Published: December 2016

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Abstract

The Cauchy problem of the 3D Zakharov-Kuznetsov equation $$ u_{t}+\partial_{\tilde{x}_{*,1}}\Delta u +(u^2)_{\tilde{x}_{*,1}}=0, (x,t)\in \mathbb{R}^3 \times \mathbb{R}, \ x=(\tilde{x}_{*,1},\tilde{x}_{*,2},\tilde{x}_{*,3});$$ is considered. It is shown that it is globally well-posed in energy space $H^1(\mathbb{R}^3)$. It answer an open problem: Is it globally well-posed in energy space $H^1 (\mathbb{R}^3)$ for 3D Z-K equtation [10,12,13]?
Moreover, in 4-D and more higher dimension, it is shown that it is locally well-posed in $H^1(\mathbb{R}^n)$ with $n\geq 4$.
The method in this paper combine the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequality). We mainly extend the spaces $\mathbf{F}^s$ and $\mathbf{N}^s$ in one dimension [4] to higher dimension.

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Mathematics Subject Classification: Primary: 35E15; Secondary: 35Q53.

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