A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results

TL;DR

This paper investigates the well-posedness, global existence, and scattering of solutions to the 2D generalized Zakharov-Kuznetsov equation, providing new results for different nonlinearities and initial data conditions.

Contribution

It establishes local well-posedness for high nonlinearities, sharp criteria for global solutions, and a scattering result under small initial data in Sobolev and Lebesgue spaces.

Findings

Local well-posedness for $k extgreater 8$ in $H^s$ spaces

Sharp global existence criteria for $k extgreater 3$ in $H^1$

Scattering results for small initial data in $H^1$

Abstract

We consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+\partial_x \Delta u+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev spaces $H^s(\mathbb{R}^2)$, where $s$ is greater than the critical scaling index $s_k=1-2/k$. For $k\geq 3$ we also establish a sharp criteria to obtain global $H^1(\R^2)$ solutions. A nonlinear scattering result in $H^1(\R^2)$ is also established assuming the initial data is small and belongs to a suitable Lebesgue space.