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The Cauchy problem for Zakharov-Kuznetsov equation on $\mathbb{R}^2$ is shownto be global well-posed for the initial date in $H^{s}$ provided$s>-\frac{1}{13}$. As conservation laws are invalid in Sobolev spaces below$L^2$, we construct an almost conserved quantity using multilinear correctionterm following the $I$-method introduced by Colliander, Keel, Staffilani,Takaoka and Tao. In contrast to KdV equation, the main difficulty is to handlethe resonant interactions which are significant due to the multidimensional andmultilinear setting of the problem. The proof relies upon the bilinearStrichartz estimate and the nonlinear Loomis-Whitney inequality.

Resonant Decompositions and Global Well-posedness for 2D Zakharov-Kuznetsov Equation in Sobolev spaces of Negative Indices