Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications

TL;DR

This paper establishes local and global well-posedness results for the Zakharov-Kuznetsov equation in various Sobolev spaces using bilinear Strichartz estimates and advanced harmonic analysis techniques.

Contribution

It introduces a bilinear Strichartz estimate in Bourgain's spaces for the Zakharov-Kuznetsov equation, enabling new well-posedness results in multiple dimensions.

Findings

Local well-posedness in H^s(R^2) for s > 1/2

Global well-posedness in H^1(R x T) and H^s(R^3) for s > 1

Use of bilinear Strichartz estimates and recent sharp Strichartz results

Abstract

This article is concerned with the Zakharov-Kuznetsov equation {equation} \label{ZK0} \partial_tu+\partial_x\Delta u+u\partial_xu=0 . {equation} We prove that the associated initial value problem is locally well-posed in $H^s(\mathbb R^2)$ for $s>\frac12$ and globally well-posed in $H^1(\mathbb R\times \mathbb T)$ and in $H^s(\R^3) $ for $ s>1$. Our main new ingredient is a bilinear Strichartz estimate in the context of Bourgain's spaces which allows to control the high-low frequency interactions appearing in the nonlinearity of \eqref{ZK0}. In the $\mathbb R^2$ case, we also need to use a recent result by Carbery, Kenig and Ziesler on sharp Strichartz estimates for homogeneous dispersive operators. Finally, to prove the global well-posedness result in $ \R^3 $, we need to use the atomic spaces introduced by Koch and Tataru.