In this paper, we prove the global well-posedness for the incompressible magnetohydrodynamics (MHD) equations in the three-dimensional unbounded domain $\mathrm{\Omega }:={\mathbb{R}}^{+}\times {\mathbb{R}}^2$. More precisely, we construct global small Sobolev regularity solutions with the initial data near 0 for the three-dimensional MHD equations in Ω. The key point of the proof is to find the suitable initial approximation function such that the linearized equations around it admitted a partial dissipative structure when we carry out the weighted energy estimate. Meanwhile, the asymptotic expansion of Sobolev regularity solutions is given.

中文翻译：

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不可压缩磁流体动力学方程的整体小有限能量解 ${\mathbb{[R}}^{+}\times {\mathbb{[R}}^2$

在本文中，我们证明了三维无界域中不可压缩磁流体动力学（MHD）方程的整体适定性 $\mathrm{\Omega }：={\mathbb{[R}}^{+}\times {\mathbb{[R}}^2$。更准确地说，对于Ω中的三维MHD方程，我们用接近0的初始数据构造全局小Sobolev正则解。证明的关键是找到合适的初始逼近函数，以便当我们进行加权能量估计时，围绕它的线性方程式允许部分耗散结构。同时给出了Sobolev正则解的渐近展开。