Global well-posedness and scattering for the 2D modified Zakharov-Kuznetsov equation

TL;DR

This paper establishes local and global well-posedness, along with scattering results, for the 2D modified Zakharov-Kuznetsov equation in a new two-parameter function space, highlighting the sharpness of these results.

Contribution

Introduces a novel two-parameter space for the 2D modified Zakharov-Kuznetsov equation and proves sharp well-posedness and scattering results within this framework.

Findings

Local well-posedness for s+a ≥ 1/4, 0 < a < 1/4

Global well-posedness and scattering for small data at s=0, a=1/4

Results are sharp in the sense of C^3-flows

Abstract

We consider the Cauchy problem associated with the modified Zakharov-Kuznetsov equation over $\mathbb{R}^2$. Taking into consideration the associated dispersive effects, we introduce, for $s,a\ge 0$, a two-parameter space $H^{s,a}(\mathbb{R}^2)$, which scales as the classic $H^s$ spaces. In this new class, we prove local well-posedness for $s+a\ge 1/4$, $0<a<1/4$, and global well-posedness and scattering for small data in the case $s=0, \ a=1/4$. These results are shown to be sharp in the sense of $C^3$-flows.