某些非线性可压缩流体方程组解的整体适定性及其吸引子

The project investigates the global well-posedness of solutions and their attractors for the following nonlinear compressible evolution equations arising from hydrodynamics: (1) global well-posedness of solutions for the 1D compressible radiative magnetohydrodynamic (MHD) equations, low Mach number limit of 1D non-isentropic compressible magnetohydrodynamic equations, and global well-posedness of solutions to 3D compressible quantum magnetohydrodynamic equations with degenerate viscosity and damping; (2)the existence and upper semicontinuity of uniform (pullback) attractors for the 1D nonlinear nonautonomous compressible radiative magnetohydrodynamic equations, and the existence and upper semicontinuity of global (uniform/pullback) attractors for the 1D nonlinear autonomous (nonautonomous) non-isentropic compressible magnetohydrodynamic equations; (3) long time behavior of solutions for the 3D compressible magneto-micropolar equations. These problems are all extremely active, challenging and important ones in recent ten years. Therefore, a deep detailed study undertaken on these problems needs creativities in both theories and applications, and even necessarily creats important impacts on the research of theories and applications for nonlinear evolutionary equations and their infinite-dimensional dynamical systems.

本项目拟研究在流体力学中出现的下列非线性可压缩发展方程组解的整体适定性及其吸引子的存在性：（1）一维带辐射的可压缩磁流体力学方程组解的整体适定性，一维非等熵可压缩磁流体力学方程组的低马赫数极限，具有退化粘性和阻尼的三维可压缩量子磁流体力学方程组解的整体适定性；（2）一维带辐射的非线性非自治可压缩磁流体力学方程组一致（拉回）吸引子的存在性和上半连续性，一维非等熵自治（非自治）可压缩磁流体力学方程组整体（一致、拉回）吸引子的存在性和上半连续性；（3) 三维可压缩磁微极流体方程组解的长时间行为。 这些问题都是近十几年在国际上极为活跃的和极具挑战性的重要问题。因此，对这些问题开展深入细致的研究，不仅需要在理论和应用上进行创新，而且也必将在非线性发展方程及无穷维动力系统的理论和应用上产生重要影响。

本项目组深入细致地研究了起源于流体力学中的几类重要的非线性发展方程组：（1）磁流体力学方程组；（2）微极流体力学方程组；（3）非牛顿流体力学方程组；（4）辐射流体力学方程组，得到了这些方程组解的整体适定性及其无穷维动力系统吸引子的存在性。这些方程组均是国际前沿的流体力学中的数学问题，在项目组成员的共同努力下，课题组发表了4篇学术论文，这些论文的结论包括整体解的存在性和渐近性，低马赫数极限以及整体吸引子的存在性等。这些结论在理论和应用上有重要的意义，对相关学科有重要的影响。