On the periodic Zakharov-Kuznetsov equation

Felipe Linares; Mahendra Panthee; Tristan Robert; Nikolay Tzvetkov

doi:10.3934/dcds.2019145

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2019, Volume 39, Issue 6: 3521-3533. Doi: 10.3934/dcds.2019145

This issue Previous Article On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities Next Article Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow

On the periodic Zakharov-Kuznetsov equation

1.
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil
2.
IMECC-UNICAMP, 13083-859, Campinas, São Paulo, Brazil
3.
Université de Cergy-Pontoise, Cergy-Pontoise, F-95000, UMR 8088 du CNRS, France

^* Corresponding author: Tristan Robert
^* Corresponding author: Tristan Robert

Received: September 2018

Revised: November 2018

Published: February 2019

FL was partially supported by FAPERJ and CNPq Brasil, MP was partially supported by FAPESP (2016/25864-6) and CNPq (305483/2014-5) Brasil

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We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on $ \mathbb T^2 $. We prove the local well-posedness for given data in $ H^s( \mathbb T^2) $ whenever $ s> 5/3 $. More importantly, we prove that this equation is of quasi-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the $ \mathbb R^2 $ and $ \mathbb R\times \mathbb T $ settings.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B05.

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