Article ID Journal Published Year Pages File Type
4604127 Annales de l'Institut Henri Poincare (C) Non Linear Analysis 2015 25 Pages PDF
Abstract

This article is concerned with the Zakharov–Kuznetsov equationequation(0.1)∂tu+∂xΔu+u∂xu=0.∂tu+∂xΔu+u∂xu=0. We prove that the associated initial value problem is locally well-posed in Hs(R2)Hs(R2) for s>12 and globally well-posed in H1(R×T)H1(R×T) and in Hs(R3)Hs(R3) for s>1s>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 R2R2 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 R3R3, we need to use the atomic spaces introduced by Koch and Tataru.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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