Abstract The Cauchy problem of the 2D Zakharov–Kuznetsov equation ∂ t ⁡ u + ∂ x ⁡ ( ∂ x ⁢ x + ∂ y ⁢ y ) ⁡ u + u ⁢ 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.

中文翻译：

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二元双线性估计及其在端点空间 𝐻−1/4 中二维 Zakharov-Kuznetsov 方程适定性的应用

摘要 二维 Zakharov–Kuznetsov 方程的柯西问题 ∂ t ⁡ u + ∂ x ⁡ ( ∂ x ⁢ x + ∂ y ⁢ y ) ⁡ u + u ⁢ ux = 0 {\partial_{t}u+\partial_{x} (\partial_{xx}+\partial_{yy})u+uu_{x}=0} 被考虑。结果表明，二维 ZK 方程在端点 Sobolev 空间 H - 1 / 4 {H^{-1/4}} 局部适定，在 H - 1 / 4 {H^ {-1/4}} 初始数据较小。在本文中，我们主要建立了一些新的二元双线性估计以获得结果，其中主要的新颖性是根据单变量多项式参数化共振函数的奇异性。