Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4604127 | Annales de l'Institut Henri Poincare (C) Non Linear Analysis | 2015 | 25 Pages |
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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Luc Molinet, Didier Pilod,