Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications
By Luc Molinet, Didier Pilod. 2013
109 citations recorded by the source corpus.
Abstract This article is concerned with the Zakharov–Kuznetsov equation (0.1) ∂ t u + ∂ x Δ u + u ∂ x u = 0 . We prove that the associated initial value problem is locally well-posed in H s ( R 2 ) for s > 1 2 and globally well-posed in H 1 ( R × 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 (0.1) . In the 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.
Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/a8bde4d0cd09578d4507e04ebec590f87449a6db
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In Applied Mathematics
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D (2019, background): “Pilod in [14] obtained independently the well-posedness of (1.”