多物理过程耦合的流体动力学方程

This project is aimed to study some difficult but fundamental mathematical problems of partial differential equations in fluid dynamics. As is well-known that, there are not only convection, viscosity and nonlinear mechanism but also dispersion and other physical mechanisms in typical nonlinear partial differential equations of fluid dynamics in multi-physics coupling processes. It focuses on some notable open problems in both mathematics and physics, such as the global well-posedness and large time behavior of smooth solutions, blow-up mechanism of solutions, existence and uniqueness of weak solutions and so on. The mathematical studies on these problems are essential for the understanding of corresponding physical processes. To solve these problems, we need not only creative mathematical methods but also comprehension of different physical mechanisms (for instance, the interactions among different physical quantities, dissipation and dispersion effects, physical and geometrical structures, etc.) to find corresponding invariants so as to formulate global dynamical behaviors of the corresponding mathematical problem. In addition to further improvement of some traditional techniques like a priori estimates based on methods of relative entropy and weak compactness, this project is also devoted to the innovation of mathematical methods applicable to the difficult problems arising in this project. We focus on the creative improvement of some modern harmonic analysis methods such as micro-local analysis, covariant derivatives and frequency localization and so on, for further understanding and solving of some difficult problems in this field.

本项目致力研究流体动力学方程一些困难的基本数学问题。众所周知,以流体动力学为主线的多物理过程耦合的一些典型偏微分方程不仅仅有对流、粘性和非线性机制的作用，同时还含有色散效应等多种物理机制,研究诸如：光滑解的整体适定性与长时间行为、解的Blow-up机制、弱解的存在性和是否是唯一的物理解等皆是数学物理界关注的公开问题。这些问题的数学研究对于理解相应的物理过程是至关重要的。解决这些问题除了深刻的数学方法之外，还需要理解这些问题的各种物理机制（诸如：各物理量之间的相互作用、耗散与色散效应、物理与几何结构等），寻求相应的不变量，以期获得相应数学问题的整体动力学行为。本项目除了进一步深化通过相对熵和弱紧性等方法建立先验估计等传统手段外，着力于数学方法的创新，以适应本项目提出的困难问题。我们着力于微局部分析、协变导数与频率局部化等现代调和分析方法的改进与创新，促使对这一领域一些困难问题的理解和解决。

以流体动力学为主线的多物理过程耦合的偏微分方程的整体适定性与长时间行为、解的Blow-up机制是数学、物理界乃至工程界关注的公开问题。 这些方程组 不仅仅有对流、粘性和非线性机制的作用，同时还含有色散效应等多种物理机制。解决这些问题除了深刻的数学方法之外，还需要理解这些问题的各种物理机制。本课题着力于数学方法的创新，特别注重微局部分析与频率局部化等现代调和分析方法的改进与创新，以及和传统能量方法、Green函数方法的改进与结合，对这一领域一些困难问题作出一系列的工作，特别是在用调和分析研究非线性偏微分方程，以及多物理过程耦合的流体力学方程研究取得一系列有意义的结果。