Scattering of the 3D Zakharov-Kuznetsov equation

TL;DR

This paper proves that solutions to the 3D Zakharov-Kuznetsov equation with small weighted initial data scatter in the energy space, using space-time resonance methods and anisotropic analysis.

Contribution

It establishes scattering for the 3D Zakharov-Kuznetsov equation with small weighted initial data, employing novel anisotropic weighted norms and resonance techniques.

Findings

Global solutions scatter in H^1 for small weighted initial data.

Dispersive decay estimates are established for anisotropic norms.

A bootstrap argument is used to prove the main scattering result.

Abstract

We consider the Zakharov-Kuznetsov equation in space dimension 3: \[ \left\{ \begin{array}{l} \partial_t u + \partial_x \Delta u + \partial_x \frac{u^2}{2} = 0 \\ u(t = 0) = u_0 \end{array} \right. \] where $u : (t, x, y) \in \mathbb{R} \times \mathbb{R} \times \mathbb{R}^2 \mapsto u(t, x, y) \in \mathbb{R}$, and $\Delta = \partial_x^2 + \Delta_y$ is the full Laplacian. We show that, for any $u_0$ satisfying \[ \Vert (1 + x^2 + |y|^2) u_0 \Vert_{H^1} \ll 1 \] then the global solution exhibits scattering in $H^1$. This is done using the method of space-time resonances, and more precisely the partial symmetries approach [GPW23] in order to treat the anisotropy. We introduce well suited anisotropic weighted norms, prove dispersive decay estimates adapted to these norms and an a priori estimate allowing to close by a bootstrap argument.