On the generalized Zakharov-Kuznetsov equation at critical regularity

TL;DR

This paper proves local and global well-posedness for the generalized Zakharov-Kuznetsov equation at critical regularity in 2D and 3D, using advanced harmonic analysis techniques and function space frameworks.

Contribution

It extends well-posedness results to the critical Sobolev space for the equation in higher dimensions, employing novel estimates and function space methods.

Findings

Establishes local well-posedness at critical regularity.

Proves global well-posedness for small initial data.

Utilizes advanced harmonic analysis tools and function space frameworks.

Abstract

The Cauchy problem for the generalized Zakharov-Kuznetsov equation $$\partial_t u +\partial_x\Delta u=\partial_x u^{k+1}, \qquad \qquad u(0)=u_0$$ is considered in space dimensions $n=2$ and $n=3$ for integer exponents $k \ge 3$. For data $u_0 \in \dot{B}^{s_c}_{2,q}$, where $1\le q \le \infty$ and $s_c=\frac{n}{2}- \frac{2}{k}$ is the critical Sobolev regularity, it is shown, that this problem is locally well-posed and globally well-posed, if the data are sufficiently small. The proof follows ideas of Kenig, Ponce, and Vega and uses estimates for the corresponding linear equation, such as local smoothing effect, Strichartz estimates, and maximal function inequalities. These are inserted into the framework of the function spaces $U^p$ and $V^p$ introduced by Koch and Tataru.