Global well-posedness and scattering for the 2D modified Zakharov-Kuznetsov equation

We consider the Cauchy problem associated with the modified Zakharov-Kuznetsov equation over ${\mathbb{R}}^2$. Taking into consideration the associated dispersive effects, we introduce, for $s,a\ge 0$, a two-parameter space $H^{s,a}({\mathbb{R}}^2)$, which scales as the classic $H^s$ spaces. In this new class, we prove local well-posedness for $s+a\ge 1\mathrm{/}4$, $0<a<1\mathrm{/}4$, and global well-posedness and scattering for small data in the case $s=0,a=1\mathrm{/}4$. These results are shown to be sharp in the sense of $C^3$-flows.