Well-posedness for the Cauchy problem of the modified Zakharov-Kuznetsov equation

Abstract

This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on ${\mathbb{R}}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s\ge 1/4$. If $d\ge 3$, by employing $U^p$ and $V^p$ spaces, we establish the small data global well-posedness in the scaling critical Sobolev space $H^{s_c}({\mathbb{R}}^d)$ where $s_c=d/2-1$.

15 pages