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 $s+a\ge 1/4$, $0<a<1/4$, and global well-posedness and scattering for small data in the case $s=0, \ a=1/4$. These results are shown to be sharp in the sense of $C^3$-flows.
Categories: math.AP