两相流稳定性的数学理论分析

In this proposal, we shall study the mathematical theory of the stability of two-phase flow. Focusing on the stability conditions of the interface, with both sides of the interface being two different fluids, such as ideal fluids and magnetohydrodynamics fluids, we discuss the well-posedness, stability and long time behavior of the whole system. The project is intended to carry out the following researches: (1) The well-posedness, stability and long time behavior of the two-phase flow system. By developing the theory of free boundaries, the harmonic analysis and micro-local analysis in fluid mechanical equations and magnetohydrodynamics equations, we aim to study the key factors for determining the stability of interface and flows, and the well-posedness and interface stability of ideal two-phase flow equations of fluid and magnetic fluid. (2) Analysis of the interface layer. By using the multi-scale analysis and establishing uniform estimate of strong solutions, we study the small viscosity limit and behavior of interface layer for the two-phase flow equations with viscous flow and magnetohydrodynamic equations. (3) Interaction between interface layer and high frequency oscillation waves. From the view of nonlinear geometric optics, we consider the effect of refraction and reflection highly oscillatory waves on the system when incoming oscillatory waves in the magnetic flow interact with the interface. The research of this project will enrich the mathematical theory of multiphase flow, free boundary problem and wave stability theory. Meanwhile, it will also push forward the mathematical theory of the nonlinear partial differential equation.

本项目将研究两相流稳定性的数学理论。重点考察界面的稳定性条件，以及界面两侧是不同介质的流体时，如理想流体和磁流体，讨论整个系统的适定性、稳定性和长时间性态。本项目拟开展以下研究：（1）两相流系统的适定性，稳定性和长时间性态。通过发展流体力学方程组的自由边界问题理论、调和分析与微局部分析工具等，揭示制约流体以及界面稳定的关键因素，考虑理想流体和磁流体的两相流系统的适定性、界面的稳定性；（2）界面层的分析。研究带粘性的流体和磁流体力学方程组的两相流系统，利用多尺度分析、强解的一致估计等考察小粘性极限，和理想流体的关系以及界面层理论；（3）界面层与高频振荡波的干扰理论。从非线性几何光学的角度，考虑当磁流体一侧的高频波与界面干扰时，反射波与折射波对系统的稳定性效应。本项目的研究将丰富多相流的数学理论，自由边界问题，以及波的稳定性理论等。同时促进非线性偏微分方程理论和方法的研究。

不同性态的两相流的稳定性和不稳定性有其重要的应用背景，也是经典的重要研究课题。本项目主要致力于流体和磁流体耦合的两相流系统的稳定性的理论研究。 针对上重下轻的两层互不相溶的不同性态的流体，通过对有粘性和无粘情况进行研究，得到了磁场和粘性对系统稳定性的影响机制；针对可压流体力学方程组边界层的光滑解的爆破问题，通过给出特殊的流体初始速度，在没有单调性假设的条件下，证明了可压Navier-Stokes方程组边界层方程的光滑解在有限时间爆破； 针对弹性流体静力学方程组的适定性问题，考虑弹性流体静力学极限方程组，没有凸性假设，要求弹性张量的切向分量不消失，证明了方程组的局部适定性，说明弹性张量对静力学极限方程组有稳定作用；针对带二次耦合非线性项的薛定谔方程组的单峰解的存在性问题，给出合适的条件，证明了方程组的单峰解的存在性。应用有限约化方法，证明了方程组存在单峰解。应用局部Pohazaev恒等式，考虑方程组单峰解的局部唯一性和多峰解的存在性。