Abstract
Abstract The Cauchy problem of the 2D ZakharovâKuznetsov equation â t ⥠u + â x ⥠( â x âą x + â y âą y ) ⥠u + u âą u x = 0 {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space H - 1 / 4 {H^{-1/4}} , and it is globally well-posed in H - 1 / 4 {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.
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