可压缩流体的瑞利-泰勒不稳定性的研究

The equations of hydrodynamics, in spite of their complexity, allow some simple patterns of flow (such as between parallel planes, or rotating cylinders) as stationary solutions. These patterns of flow can, however, be realized only for certain ranges of the parameters characterizing them. Outside these ranges, they cannot be realized. The reason for this lies in their inherent instability, i.e. in their inability to sustain themselves against small perturbations to which any physical system is subject. One important case of this instability is Rayleigh-Taylor instability which is an acceleration-driven instability of density inverted fluid interfaces. And the above instability is abundant in nature, for example, oceans, atmospheres and plasma. The classical study deals with linear incompressible fluid model. While the main focus of this program is on the effects of compressibility as well as entropy on the nonlinear instability. We will show that even if a compressible system has no density inversion anywhere, the system is still unstable. Our research mainly includes four aspects: single layer fluids, multi-layer fluids, viscous or rotating fluids and electrical fluids subjected to magnetic field. We hope to know how the compressibility as well as entropy impact the Rayleigh-Taylor instability exactly.

流体动力学方程组存在一些简单的静态解，这些静态解仅在系统各参数的小幅度变化范围内存在。一旦超过这些范围，就不存在上述的静态解了。究其原因，这些静态解本身无法抵抗任何小的扰动，即系统与生俱来的不稳定性导致的。这其中就包含一种重要的不稳定性，即由惯性力驱动的Rayleigh-Taylor不稳定性。这一不稳定性广泛的存在于大气，海洋，等离子体等自然现象当中。以往的研究多集中在不可压流体的线性模型，本项目集中研究可压缩性以及熵对该非线性不稳定性的影响，试图证明可压缩系统比不可压缩系统更不稳定。甚至即便系统流体密度上轻下重，依然存在Rayleigh-Taylor不稳定。我们准备由浅入深的分四个层次展开研究，单层可压流体，多层可压流体，带粘性或旋转的可压流体以及磁场中的带电可压流体，希望能把可压缩性以及熵对流体Rayleigh-Taylor不稳定性的影响研究清楚。

本项目主要从动力系统角度，研究变密度粘性流体动力学中非线性瑞利—泰勒不稳定性的数学理论，其中包括可压流体、非齐次不可压磁流体、以及对应的分层流等模型。所采用的证明框架为：首先建立模型解的局部适定性；其次构造对应的线性化方程组的增长性解及线性算子的谱；最后把非线性方程组写成温和解的形式，使得主要的线性部分能够控制非线性部分，从而得出非线性不稳定性。由于这些模型都有各自的特点，为了克服其所产生的数学难点，我们探索了一些新的变换方法及一般的变分方法来研究密度连续情况模型，然后进一步研究分层流中的自由边界问题。我们的结果不但丰富了流体动力学中有关不稳定性的数学理论，而且还为实际物理应用提供数学角度的理论指导。