Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications

Abstract

This article is concerned with the Zakharov-Kuznetsov equation {equation} \label{ZK0} \partial_tu+\partial_xΔu+u\partial_xu=0 . {equation} We prove that the associated initial value problem is locally well-posed in $H^s({\mathbb{R}}^2)$ for $s>\frac{1}{2}$ and globally well-posed in $H^1(\mathbb{R}\times \mathbb{T})$ and in $H^s({\mathbb{R}}^3)$ 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 ${\mathbb{R}}^3$, we need to use the atomic spaces introduced by Koch and Tataru.

25 pages; in this new version, we also proved global well-posedness in $H^1(R\times T)$ and in $H^s(R^3)$, for $s>1$

References in corpus (1)

• Global well-posedness for the KP-I equation on the background of a non localized solution