发表机构
Institute of Applied Mathematics & State Key Laboratory of Mathematical Sciences, AMSS, & Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences; School of Mathematics and Computer Sciences, Institute of Mathematics and Interdisciplinary Sciences, Nanchang University; Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所; 南昌大学数学与计算机科学学院; 中国科学院数学与系统科学研究院应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究无界域上一维可压缩MHD方程全局强解的大时间行为，通过构造辅助椭圆方程得到新估计，证明磁场偏差有界、比容和温度有界，且解渐近稳定。
AI 中文摘要
本文研究了无界域上一维平面可压缩磁流体动力学（MHD）系统全局强解的大时间行为。采用正常数粘性系数，热导率与温度的非负次幂成正比。考虑了整条实直线上的柯西问题以及半直线上的两个初边值问题。对于这些半直线问题，边界是热绝缘的，并对磁场施加齐次诺伊曼或齐次狄利克雷边界条件。通过关键构造一个辅助椭圆方程，从MHD系统的能量密度方程推导出一个新的估计。该估计表明，磁场强度的平方与其空间平均值之间的偏差具有时间平方根界，这进而意味着比容在空间和时间上具有一致的上界和正下界。此外，还建立了温度的一致时间上下界，并证明了当时间趋于无穷时，全局解是渐近稳定的。
英文摘要
Global existence and the large-time behavior of strong solutions to the one-dimensional planar compressible magnetohydrodynamic system in unbounded domains are investigated.The longitudinal viscosity is assumed to be a positive constant plus any nonnegative power of the density, or to depend on temperature through a power law. The heat conductivity is taken to be proportional to any nonnegative power of temperature, while the transverse viscosity and magnetic diffusivity are taken to be positive constants. Global existence, uniform-in-time estimates, and convergence to equilibrium are established without imposing any smallness conditions on the initial data. In the density-dependent case, any fixed nonnegative viscosity exponent is allowed; in the temperature-dependent case, the nonnegative viscosity exponent is required to be sufficiently small in terms of the initial data. An auxiliary elliptic equation is introduced to control magnetic-pressure oscillations through a square-root-in-time estimate. Combined with the estimates appropriate to each viscosity law, this yields uniform upper and positive lower bounds for the specific volume and temperature and asymptotic stability of the global strong solution.
CommentsRevised version. The main results have been strengthened to cover a broader physically relevant viscosity regime
