粘性流体力学的大初值问题研究

粘性流体力学方程组是数学流体力学中的一类非常重要的模型，除了在基本的流体力学中，还在海洋运动、大气物理等理论及工程等方面具有广泛的应用。然而在高维情形关于粘性流体的大初值问题研究较少。. 本项目拟研究可压缩NS方程组特殊大解的构造、在无界区域中可压缩流体大解的长时间行为和稳定性、可压缩NS方程在薄区域中的适定性和极限。本项目将引入慢变空间变量来构造一系列特殊大解；深入分析方程组算子谱各频率的不同行为，通过它们之间的平衡关系来克服大初值带来的困难；通过刻画密度下界的发展情况来避免可能出现的退化情况，并结合方程内在的抵消机制与调和分析工具进行研究。. 本项目的研究有助于深刻理解粘性流体大初值、算子谱和内在结构之间的相互联系，为探索流体力学奥秘、物理实验及应用提供数学理论指导。

Viscous fluid dynamical system is a very important model in mathematical fluid mechanics. Beyond the classical fluid mechanics, it has a wide range of application in oceanography, atmosphere physics and other theoretical and engineering aspects. However, in the multi-dimensional case, the large initial data problem for viscous fluid has less understanding and investigation...We will study in this project the construction of large solution to compressible NS equations, the long-time behavior and stability of large solution to compressible fluids in unbounded domain, and well-posedness and limit of compressible NS equations in the thin domain. To do it, we will introduce the slow varying space variable to construct a series of large solutions, deeply analyze the different asymptotic behavior of low and high spectrum, utilize the balance between them to overcome the difficulties caused by large initial data, describe the development of low bound of density to avoid the possible degeneration, which will be combined with the use of cancellation and harmonic tools...The research project can help us to understand the relations between the large initial data, the spectrum and the inner construction of equations, and provide mathematical theoretical guidance for exploring the mysteries of fluid mechanics, physical experiments and applications.