In this paper, we study the global existence and asymptotic dynamics of generalized magnetohydrodynamic equations in \({\mathbb {R}}^3\), in which the dissipation terms are \(-\eta (-\Delta )^\alpha \) and \(-\mu (-\Delta )^\beta \), \(0<\alpha ,\,\beta <1\). With the help of combining the local existence and the a priori estimates, we establish the global existence and uniqueness of solution with small initial data. Moreover, we obtain the asymptotic decay rates of solutions by the method of energy estimates.

中文翻译：

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3D广义磁流体动力学方程的整体存在性和渐近稳定性

在本文中，我们研究了\（{\ mathbb {R}} ^ 3 \）中广义磁流体动力学方程的整体存在性和渐近动力学，其中耗散项为\（-\ eta（-\ Delta）^ \ alpha \）和\（-\ mu（-\ Delta）^ \ beta \），\（0 <\ alpha，\，\ beta <1 \）。借助于结合局部存在和先验估计，我们建立了具有少量初始数据的整体存在性和解的唯一性。此外，我们通过能量估计的方法获得了溶液的渐近衰减率。