Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space

TL;DR

This paper establishes the well-posedness of the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev spaces, extending previous results by incorporating dissipative effects and Fourier restriction techniques.

Contribution

It proves local and global well-posedness for the 2D ZKB equation in Sobolev spaces with regularity index greater than -1/2, using Fourier restriction norms with dissipation.

Findings

Well-posedness in $H^s( ^2)$ for $s > -1/2$

Extension of well-posedness to lower regularity spaces

Use of Fourier restriction norm with dissipative effect

Abstract

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term $-\partial_x^2u$. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in $H^s(\mathbb{R}^2)$ for $s>1/2$, and the 2D nonlinear parabolic equation with quadratic derivative nonlinearity is well-posed in $H^s(\mathbb{R}^2)$ for $s\ge 0$. By using the Fourier restriction norm with dissipative effect, we prove the well-posedness for ZKB equation in $H^s(\mathbb{R}^2)$ for $s>-1/2$.