平面可压缩辐射磁流体动力学方程组Cauchy问题整体解的大时间行为

Large-Time Behavior of the Global Solution of the Cauchy Problem of the Planar Compressible Radiation Magnetohydrodynamic Equations

摘要: 关于具有大初始值的平面可压缩磁流体动力学方程柯西问题整体解的存在性已有不少结果，然而很少有结果集中在其大时间行为上，并且对于辐射磁流体方程组的动力学行为的相关研究更少. 因此，本文的主要目的是研究具有大初始值的平面可压缩辐射磁流体动力学方程组柯西问题整体解的存在性及大时间渐近行为，并考虑粘性系数是常数. 我们分析的关键点是推导出比容和绝对温度的与时间无关的一致正上下界. 本文研究的主要困难包括流体动力学和磁动力学之间的非线性耦合效应、区域的无界性以及辐射项的强非线性，在这里我们使用截断函数和能量估计方法巧妙地解决了上述困难。

Abstract: There have been many results on the existence of the global solution of the Cauchy problem of the planar compressible magnetohydrodynamic equation with large initial values. However, few results focus on its large time behavior, and there are fewer relevant studies on radiative magnetohydrodynamics. The main purpose of this paper is to study the large time behavior of the global solution of the Cauchy problem of the above system, where the viscosity coefficient is a constant. The key point of our analysis is to derive the uniform-in-time positive upper and lower bounds of specific volume and absolute temperature. This paper involves many difficulties, including the nonlinear coupling effect between fluid dynamics and magnetodynamics, the unbounded computational domain, and the strong nonlinearity of the radiation term, which we cleverly overcome by using the cut-off function and energy estimation method.