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 R2 is shown to be global well-posed for the initial data in Hs provided s>−113. As conservation laws are invalid in Sobolev spaces below L2, we construct an almost conserved quantity by multilinear correction term following the I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In contrast to the KdV equation, the main difficulty is to handle the resonant interactions due to the multidimensional and multilinear setting of the problem. The proof relies upon the bilinear Strichartz estimate and the nonlinear Loomis-Whitney inequality.