Well-Posedness for the Two-Dimensional Modified Zakharov-Kuznetsov Equation

We prove that the initial value problem for the two-dimensional modified Zakharov–Kuznetsov equation is locally well-posed for data in $H^s(\mathbb{R}^2)$, $s>3/4$. Even though the critical space for this equation is $L^2(\mathbb{R}^2)$, we prove that well-posedness is not possible in such space. Global well-posedness and a sharp maximal function estimate are also established.