流动的稳定性与不稳定性的数学理论

This project is devoted to studying the mathematical problems of RT instability, magnetic buoyancy instability, thermal convection instability and gravitation instability from the point of view of dynamical system, which includes the RT problem of stratified viscoelastic fluids, the Parker problem arising from magnetohydrodynamic fluids, the magnetic RB problem and the stability problem of steady rotation solutions of gaseous star with inner core. To solve those problem, we will develop some new mathematical method, such as the method of Bogovskii function extracting the stability condition, the modified variational method with reciprocal form, the “bootstrap” instability with weight, and so on. In addition, we will also use the latest mathematical tools, such as the Monge-Ampere theory applied to modify the initial data of viscoelastic fluids, the two-tier energy method, and the energy-Casimir energy technique with rotation, and so on. Our research methods and results not only enrich the stability and instability of the mathematical theory in the fluid dynamics, but also provide theoretical guidance in mathematical point for the physical applications. In addition, we will study the spectrum problem of the first three problems under the framework of Sobolev spaces, it will also enrich and perfect the spectrum theory in fluid dynamics.

本项目主要利用流体力学方程组，从动力系统角度研究与RT不稳定、磁浮力不稳定、热对流不稳定以及引力不稳定相关的数学问题，包括有关分层粘弹性流体的RT问题，磁流体的Parker问题，磁RB对流问题，以及带有固体核的星云模型的旋转稳态解的稳定性问题。为了解决这些问题，本项目将会提出一些新的数学方法，如Bogovskii函数提炼稳定性条件方法、加权“脱靴”不稳定性方法以及反比形式的修正变分法等。此外，本项目还将引入一些新的数学工具，如用于修正粘弹性流体初始条件的Monge-Ampere方程理论、两系能量方法以及带旋转效应的Casimir能量泛函等。本项目的研究方法与结果不但会丰富流体动力学中有关稳定与不稳定性的数学理论，而且还能为实际物理应用提供数学角度的理论指导。此外，本项目将在Sobolev空间框架下研究前三个问题对应的谱问题，这将进一步丰富并完善有关流体动力学的谱理论。

本项目主要得到如下结果：1.建立起不可压缩/可压缩的分层黏弹性流体的瑞利-泰勒问题的稳定性与不稳定性的数学结果；2.建立起无磁耗散的可压缩磁流体的帕克问题的稳定与不稳定性的数学结果；3.建立起无磁耗散的不可压缩磁流体的瑞利-贝纳德问题的稳定性与不稳定性的数学结果；4.建立起分层无磁耗散的可压缩磁流体的瑞利-泰勒问题问题的稳定性与不稳定性的数学结果；5.建立起磁流体中磁场抑制流动不稳定性现象的数学物理机制。此外，我们也在其它与流体力学稳定性相关的数学问题上取得了若干进展性结果。在项目执行期间，共在《Arch. Rational Mech. Anal.》、《Calc. Var. PartialDifffferential Equations》、《J. Funct. Anal.》、《J. Math. Pures Appl.》、 《SIAM J. Math. Anal.》、《J. Difffferential Equations》、《J. London Math. Soc.》、《Proc. Roy. Soc. Edinburgh Sect. A》、《Math. Models Methods Appl. Sci.》、《J. Math. Fluid Mech.》、《Nonlinearity》、《Phys. D》等SCI数学期刊上发表论文37篇。