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We consider the modified Zakharov-Kuznetsov (mZK) equation in two spacedimensions in both focusing and defocusing cases. Using the $I$-method, weprove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ forany data in the defocusing case and under the assumption that the mass of theinitial data is less than the mass of the ground state solution of $\Delta\varphi - \varphi + \varphi^3 = 0$ in the focusing case. This improves theglobal well-posedness result of Linares and Pastor [20].

Global well-posedness for low regularity data in the 2d modified Zakharov-Kuznetsov equation