Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation

Raphaël Côte; Frédéric Valet

doi:10.3934/cpaa.2021005

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2021, Volume 20, Issue 3: 1039-1058. Doi: 10.3934/cpaa.2021005

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Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation

Raphaël Côte and
Frédéric Valet^,

IRMA UMR 7501, Université de Strasbourg, CNRS, F-67000 Strasbourg, France

^* Corresponding author
^* Corresponding author

Received: January 1, 2020

Revised: October 1, 2020

Early access: January 20, 2021

Published online: January 20, 2021

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Abstract

We consider the Zakharov-Kuznetsov equation (ZK) in space dimension 2. Solutions $ u $ with initial data $ u_0 \in H^s $ are known to be global if $ s \ge 1 $. We prove that for any integer $ s \ge 2 $, $ \| u(t) \|_{H^s} $ grows at most polynomially in $ t $ for large times $ t $. This result is related to wave turbulence and how a solution of (ZK) can move energy to high frequencies.

It is inspired by analoguous results by Staffilani [21] on the non linear Schrödinger and Korteweg-de-Vries equation. The main ingredients are adequate bilinear estimates in the context of Bourgain's spaces and a careful study of the variation of the $ H^s $ norm.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B65, 35Q35.

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