The Zakharov\u2013Kuznetsov equation in high dimensions: small initial data of critical regularity

Sebastian Herr,Shinya Kinoshita

doi:10.1007/s00028-021-00671-9

Abstract

The Zakharov–Kuznetsov equation in spatial dimension dge 5 is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces, and it is proved that solutions scatter to free solutions as t rightarrow pm infty . The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the (d-1)-dimensional Schrödinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension d=4.

Highlights

This paper is concerned with the Zakharov–Kuznetsov equation∂t u + ∂x1 u = ∂x1 u2 inR × result holds in Hsc (Rd) u(0, ·) = u0 on Rd (1.1)where d ≥ 2, u = u(t, x), (t, x) = (t, x1, . . . , xd ) ∈ R × Rd, u is real-valued, and denotes the Laplacian with respect to x.The Zakharov–Kuznetsov equation was introduced in [13] as a model for propagation of ion-sound waves in magnetic fields
The Zakharov–Kuznetsov equation can be seen as a multidimensional extension of the well-known Korteweg–de Vries (KdV) equation
We address the problem of global well-posedness and scattering for small initial data in critical spaces

Summary

Introduction

We address the problem of global well-posedness and scattering for small initial data in critical spaces. For d = 5, the Cauchy problem (1.1) is globally well-posed for small initial data in B2sc,1(R5), and solutions scatter as t → ±∞. For d ≥ 6, the Cauchy problem (1.1) is globally well-posed for small initial data in H sc (Rd ), and solutions scatter as t → ±∞. If we restrict to initial data which is radial in the last (d − 1) variables (see below for definitions), we obtain small data global well-posedness and scattering in the critical Sobolev spaces for any dimension d ≥ 4. For d ≥ 4, the Cauchy problem (1.1) is globally well-posed for small data in Hrsacd(Rd ), and solutions scatter as t → ±∞.

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