We consider the Zakharov–Kuznetsov-type limit for ion dynamics system in ${\mathbb{R}}^3$ with external magnetic field and derive the 3D ZK-type equation from ion dynamics system by Gardner–Morikawa transforms. By making energy estimate for the remainders system we prove that the solutions of the Euler–Poisson system converge globally to ZK-type equation when $\epsilon \to 0$ in 3D.

中文翻译：

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带有外部磁场的离子动力学系统的Zakharov–Kuznetsov型极限 ${\mathbb{[R}}^3$

我们考虑Zakharov–Kuznetsov型离子动力学系统的极限 ${\mathbb{[R}}^3$在外部磁场作用下，通过Gardner–Morikawa变换从离子动力学系统推导出3D ZK型方程。通过对余数系统进行能量估计，我们证明了Euler-Poisson系统的解在以下情况下全局收敛于ZK型方程$\epsilon \to 0$ 在3D中。