The    existence   of strong  solutions     to the $3D$ Zakharov-Kuznestov equation in a bounded domain

Chuntian Wang

doi:10.3934/dcds.2014.34.4897

Article Contents

2014, Volume 34, Issue 11: 4897-4910. Doi: 10.3934/dcds.2014.34.4897

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The existence of strong solutions to the $3D$ Zakharov-Kuznestov equation in a bounded domain

Chuntian Wang^1,

1.
Department of Mathematics and The Institute, for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405-5701

Received: September 30, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

We consider the Zakharov-Kuznestov (ZK) equation posed in a limited domain $\mathcal{M}=(0,1)_{x}\times(-\pi /2, \pi /2)^d,$ $ d=1,2$ supplemented with suitable boundary conditions. We prove that there exists a solution $u \in \mathcal C ([0, T]; H^1(\mathcal{M})) $ to the initial and boundary value problem for the ZK equation in both dimensions $2$ and $3$ for every $T>0$. To the best of our knowledge, this is the first result of the global existence of strong solutions for the ZK equation in $3D$.
More importantly, the idea behind the application of anisotropic estimation to cancel the nonlinear term, we believe, is not only suited for this model but can also be applied to other nonlinear equations with similar structures.
At the same time, the uniqueness of solutions is still open in $2D$ and $3D$ due to the partially hyperbolic feature of the model.

Keywords:

Zakharov-Kuznetsov equation,
Korteweg-de Vries equation.

Mathematics Subject Classification: Primary: 35A01; Secondary: 35Q35.

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