Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications

Abstract

This article is concerned with the Zakharov–Kuznetsov equation(0.1)∂tu+∂xΔu+u∂xu=0. We prove that the associated initial value problem is locally well-posed in Hs(R2) for s>12 and globally well-posed in H1(R×T) and in Hs(R3) 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 R2 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 R3, we need to use the atomic spaces introduced by Koch and Tataru.