• Corpus ID: 210911809

# Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher

@article{Herr2020SubcriticalWR,
title={Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher},
author={Sebastian Herr and Shinya Kinoshita},
journal={arXiv: Analysis of PDEs},
year={2020}
}
• Published 24 January 2020
• Mathematics
• arXiv: Analysis of PDEs
The Zakharov-Kuznetsov equation in space dimension $d\geq 3$ is considered. It is proved that the Cauchy problem is locally well-posed in $H^s(\mathbb{R}^d)$ in the full subcritical range $s>(d-4)/2$, which is optimal up to the endpoint. As a corollary, global well-posedness in $L^2(\mathbb{R}^3)$ and, under a smallness condition, in $H^1(\mathbb{R}^4)$, follow.

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• Mathematics
SIAM J. Math. Anal.
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It is proved that the local well-posedness of the three-dimensional Zakharov--Kuznetsov equation $\partial_tu+Delta\partial_xu+ u\partial-xu=0$ in the Sobolev spaces and in the Besov space.

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This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the

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