A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results

We consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+{\mathrm{\partial }}_x\mathrm{\Delta }u+{\mathrm{\partial }}_x(u^{k+1})=0$, where $k\ge 3$ is an integer number. For $k\ge 8$ we prove local well-posedness in the $L^2$-based Sobolev spaces $H^s({\mathbb{R}}^2)$, where $s$ is greater than the critical scaling index $s_k=1-2\mathrm{/}k$. For $k\ge 3$ we also establish a sharp criteria to obtain global $H^1({\mathbb{R}}^2)$ solutions. A nonlinear scattering result in $H^1({\mathbb{R}}^2)$ is also established assuming the initial data is small and belongs to a suitable Lebesgue space.