# Global well-posedness for low regularity data in the 2d modified Zakharov-Kuznetsov equation

## TL;DR

This paper proves global well-posedness for the 2D modified Zakharov-Kuznetsov equation with low regularity initial data using the $I$-method, extending previous results to broader function spaces and initial conditions.

## Contribution

It establishes global well-posedness for $H^s$ solutions with $s > 3/4$ in the defocusing case and under mass constraints in the focusing case, improving prior results.

## Key findings

- Global well-posedness for $s > 3/4$ in the defocusing case.
- Global well-posedness under mass constraints in the focusing case.
- Extension of previous results by Linares and Pastor.

## Abstract

We consider the modified Zakharov-Kuznetsov (mZK) equation in two space dimensions in both focusing and defocusing cases. Using the $I$-method, we prove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ for any data in the defocusing case and under the assumption that the mass of the initial data is less than the mass of the ground state solution of $\Delta \varphi - \varphi + \varphi^3 = 0$ in the focusing case. This improves the global well-posedness result of Linares and Pastor [20].