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

Abstract

The Cauchy problem for Zakharov-Kuznetsov equation on $\mathbb{R}^2$ is shown to 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 correction term following the $I$-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In contrast to KdV equation, the main difficulty is to handle the resonant interactions which are significant due to the multidimensional and multilinear setting of the problem. The proof relies upon the bilinear Strichartz estimate and the nonlinear Loomis-Whitney inequality.