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 2D 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 $L^2({\mathbb{R}}^2)$.