Global Well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D

Abstract

This paper is concerned with the Cauchy problem of the $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s>-1/4$, and these are optimal up to the endpoint. We utilize the nonlinear version of the classical Loomis-Whitney inequality and develop an almost orthogonal decomposition of the set of resonant frequencies. As a corollary, we obtain global well-posedness in $L^2({\mathbb{R}}^2)$.

arXiv comment59 pages, to appear, Annales de l'Institut Henri Poincaré / Analyse non linéaire

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Cited by in corpus (7)

• Numerical study of Zakharov-Kuznetsov equations in two dimensions
• Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher
• Asymptotic K-soliton-like Solutions of the Zakharov-Kuznetsov type equations
• On the propagation of regularity for solutions of the Zakharov-Kuznetsov equation
• Dispersive Blow-up for Solutions of the Zakharov-Kuznetsov equation
• Center stable manifolds around line solitary waves of the Zakharov--Kuznetsov equation with critical speed
• The Cauchy problem for the $L^2-$critical generalized Zakharov-Kuznetsov equation in dimension 3