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Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher. (Résultats sous-critiques sur le caractère bien posé de l’équation de Zakharov-Kuznetsov en dimension supérieure ou égale à trois.) (English. French summary) Zbl 1515.35249

Summary: The Zakharov-Kuznetsov equation in space dimension \(d\geq 3\) is considered. It is proved that the Cauchy problem is locally well-posed in \(H^s(\mathbb{R}^d)\) in the full subcritical range \(s>(d-4)/2\), which is optimal up to the endpoint. As a corollary, global well-posedness in \(L^2(\mathbb{R}^3)\) and, under a smallness condition, in \(H^1(\mathbb{R}^4)\), follow.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
42B25 Maximal functions, Littlewood-Paley theory
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