Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications

Abstract

This article is concerned with the Zakharov–Kuznetsov 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 (0.1). 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.