This paper is devoted to the study of dynamical behavior near explicit finite time blowup solutions for three-dimensional incompressible Magnetohydrodynamics (MHD) equations. The global infinite energy solution near a constant vector of three-dimensional incompressible MHD equations without magnetic diffusion has been obtained by Deng–Zhang (Arch Rational Mech Anal 230:1017–1102, 2018). More precisely, we find a family of explicit finite time blowup solutions that admitted smooth initial data and infinite energy in whole space \({\mathbb {R}}^3\). After that, we prove asymptotic stability of those explicit finite time blowup solutions for 3D incompressible Magnetohydrodynamics equations in a smooth bounded domain \(\Omega _{t}:=\Big \{(t,x_1,x_2,x_3):0\le x_i\le \sqrt{\overline{T}^*-t},\quad t\in (0,\overline{T}^*),\quad i=1,2,3\Big \},\) where \(\overline{T}^*\) denotes the blowup time. This means we construct a family of stable blowup solutions for 3D incompressible Magnetohydrodynamics equations with smooth initial data in \(\Omega _t\). Meanwhile, our result can be extended to the whole space \({\mathbb {R}}^3\) by means of the same method.

本文致力于研究三维不可压缩磁流体动力学 (MHD) 方程的显式有限时间膨胀解附近的动力学行为。Deng-Zhang (Arch Rational Mech Anal 230:1017–1102, 2018) 获得了无磁扩散的三维不可压缩 MHD 方程的常数向量附近的全局无限能量解。更准确地说，我们找到了一系列明确的有限时间爆炸解决方案，它们承认整个空间\({\mathbb {R}}^3\) 中的平滑初始数据和无限能量。之后，我们证明了光滑有界域中 3D 不可压缩磁流体动力学方程的那些显式有限时间膨胀解的渐近稳定性\(\Omega _{t}:=\Big \{(t,x_1,x_2,x_3):0\le x_i\le \sqrt{\overline{T}^*-t},\quad t\in ( 0,\overline{T}^*),\quad i=1,2,3\Big \},\)其中\(\overline{T}^*\)表示爆破时间。这意味着我们用\(\Omega _t\) 中的平滑初始数据为 3D 不可压缩磁流体动力学方程构建了一系列稳定的膨胀解。同时，我们的结果可以通过相同的方法扩展到整个空间\({\mathbb {R}}^3\)。