可压流体方程自由边界问题解的适定性与稳定性研究

The free boundary problem of compressible fluid equation is one important part of fluid equation research. It has strong physical and applied background and is one of the frontier topics in the field of partial differential equation. This project mainly focus on the well-posedness, stability and large-time behavior of solutions to the free boundary problem of compressible fluid equations and related models. First, we consider free boundary problem of compressible Navier-Stokes equations in multi-dimension. For general initial data with vacuum which degenerate on the boundary, we investigate the local well-posedness and global existence of weak solutions and small perturbation solutions. Then, we study Navier-Stokes-Poisson system with free boundary which models the motion of gaseous star in the following two aspects. For isentropic case with adiabatic exponent less than critical exponent, we consider the regularity and large-time behavior of spherically symmetric solutions. For isentropic asymmetric case and non-isentropic model, we focus on the well-posedness, stability/instability of solutions. The last topic of this project is the free boundary problem of two-phase flow model. Under the Navier-slip boundary condition we study the stability/instability and dynamic behavior of steady-state Couette flows. The research on the free boundary problem in this project will not only enrich the theoretical achievements in the field of fluid research, but also reflect profound physical phenomena, which has important research value.

可压缩流体方程自由边界问题是流体方程研究的重要组成部分，有极强的物理和应用背景，是偏微分方程研究领域前沿课题之一。本项目拟围绕可压流体方程及相关模型自由边界问题解的适定性、稳定性及大时间行为展开研究，主要包括：高维可压Navier-Stokes方程自由边界问题，对一般含真空(边界退化)初值，研究解的局部适定性、弱解及小扰动解整体存在性问题；气态星体模型(Navier-Stokes-Poisson)，一方面研究等熵情形绝热指数小于临界指标时球对称解的正则性及大时间行为，另一方面研究等熵非对称情形及非等熵模型解的适定性、稳定性/不稳定性；两相流自由边界问题，研究滑移边界条件下，稳态解Couette流的稳定性/不稳定性及动力学行为。本项目流体自由边界问题的研究工作，不仅将丰富流体研究领域的理论成果，而且会反映深刻的物理现象，有重要的研究价值。

本项目主要研究了可压缩流体方程及相关模型的自由边界问题，研究了二维可压缩Navier-Stokes方程真空自由边界问题，给出了局部解的先验估计。考虑磁场作用下的自由边界问题，得到了平面磁场下解的整体存在性及流体区域的扩张速率。研究大气原始方程自由边界得到了局部解的适定性。此外，还研究了两相流模型Couette流的稳定性问题，结果正在整理。