Scattering of the 3D Zakharov-Kuznetsov equation

Abstract

We consider the Zakharov-Kuznetsov equation in space dimension 3: \[ \left\{ \begin{array}{l} \partial_t u + \partial_x Δu + \partial_x \frac{u^2}{2} = 0 \\ u(t = 0) = u_0 \end{array} \right. \] 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.

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