非线性可压缩磁流体力学方程组整体适定性及其吸引子

The project will investigate the global well-posedness and their attractors for the following nonlinear compressible magnetohydrodynamic (MHD) equations: (i) large-time behavior of solutions for 1D compressible magnetohydrodynamic(MHD) equations; (ii) the existence of universal (uniform) attractors for 1D nonlinear autonomous (non-autonomous) compressible radiative magnetohydrodynamic (MHD) equations. These problems have become extremely active and challenging in the last decade,some of which are just at the beginning stage, and incomplete in the theory in the world. In this project, we shall fully make use of the intersection of knowledges of the three aspects of the nonlinear functional analysis, nonlinear partial differential equations and infinite dimensional dynamical systems to study these problems, further reveal the structure and properties of the solutions of these concrete problems, and expect that these properties can be applied to the related problems.

本项目拟研究下列非线性可压缩磁流体力学（MHD）方程组解的整体适定性及其吸引子的存在性：（1）一维具有自重和辐射的可压缩磁流体力学（MHD）方程组解的大时间性态；（2）一维带辐射的非线性自治（非自治）可压缩MHD方程组整体（一致）吸引子的存在性。 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，理论上还不尽完善。在本项目中，申请者充分利用非线性泛函分析、非线性偏微分方程和无穷维动力系统三个方面知识的交叉和融合来研究上述问题，进一步揭示出蕴含在这些具体问题中的解的结构和性质，期望将这些性质应用到其他相关问题中。

本项目研究了下列非线性发展方程（组）解的适定性和吸引子的存在性：（1）一维带辐射的非线性自治可压缩MHD方程组整体吸引子的存在性；（2）具有二声的拟线性热扩散方程组解的整体存在性和指数稳定性；（3）强阻尼可拉伸梁方程解的一致衰减性。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，理论上还不尽完善。因此，对这些问题进行深入细致的研究，将在理论上和应用上具有重要意义。