A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations

Abstract

In this note we study the generalized 2D Zakharov-Kuznetsov equations ${\partial }_{t}u+\Delta {\partial }_{x}u+u^k{\partial }_{x}u=0$ for $k\ge 2$. By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces $H^s({\mathbb{R}}^2)$ for $s>1/4$ if $k=2$, $s>5/12$ if $k=3$ and $s>1-2/k$ if $k\ge 4$.