Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications

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 $s>\frac12$ 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.

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