气体动力学中的偏微分方程

This research project is mainly devoted to consider two important physical phenomena arising from compressible fluid dynamics: vacuum phenomenon and non-isentropic multidimensional continuous transonic flow. Mathematically, theses physical phenomena come down to studying the well-posedness of degeneate type and mixed type partial differential equations or sysyems with physical initial-boundary value conditions. The research on these problems not only has strong physical background, but also is one of the key research areas in the nonlinear partial differential equations and their applications. Until now, although there are many achivements on the researches of these problems, there is a long run to establish the complete theorey on these problems. Theorefore, based on the previous works, our research project mainly focuses on the following concrete topics: (1) The local-in-time well-posedness and conormal regularity for the quasilinear degenerate hyperbolic differential equations with discontinous initial data; (2) Global well-posedness and the asymptotic behavior of the supersonic flow with a vacuum state in an infinitely long divergent nozzle; (3) The stability of vacuum surface for the stationary supersonic flow past a large curved body (4) The well-posedness of mixed type partial differetnial equations and its application to multidimensional continous transonic problem of non-isentropic polytropic gases; (5) Conormal regularity of the solution to the compressible Euler system with vaccum boundary.

本申请项目主要探讨可压缩理想流体中两类重要的物理现象：真空问题及非等熵高维连续跨音速流。数学理论上，这些物理现象可归结为具有物理初边值条件的退化型及混合型偏微分方程的适定性研究。此类问题的研究不但具有强烈的力学背景，而且也是非线性偏微分方程理论及其应用领域所关注的重要研究方向。目前，此类问题取得了许多研究成果，但其理论上的系统性及应用上的合理性还不完善。基于前人以及我们以往的工作，本项目将重点关注下列几类问题：（1）具有间断初值的拟线性退化型双曲型方程（组）解的局部适定型及解的奇性结构；（2)张口管道中渐近趋向真空的超音速流问题解的整体适定性；（3）定常超音速流经过大弯曲物体时真空界面的稳定性（4）混合型方程适定性的研究及其在非等熵高维连续跨音速流中的应用；（5）可压缩 Euler 方程组在真空边界附近解的余法正则性刻划。

在项目的执行过程中，我们主要关注来自理想可压和不可压流体的一些偏微分方程的理论问题。此类问题可归结为具有物理初边值条件的退化型及混合型偏微分方程的适定性问题。这类问题不但具有强烈的力学背景，而且也是非线性偏微分方程理论及其应用领域所关注的重要研究方向。在项目的资助下，我们主要建立了以下几个方面的结果: (1) 高维定常超音速锥状激波的整体稳定性(Comm.Math.Phys,2014); (2) 高维非定常piston问题的局部及整体稳定性(SIAM J.Math.Anal., to appear); (3) 具有Neumann边值条件的三维外区域拟线性双曲方程的初边值问题的整体适定性(J. Differential Equations, online); (4) 螺旋对称的三维不可压Euler方程组弱解的整体存在性(J. Differential Equations, 2017)。