管道中欧拉-泊松流体的适定性问题

The research on partial differential equations is one of the core areas of mathematics. Particularly, the theory on the well-posedness issue(existence, uniqueness and regularity) for the topic of conservation law equations are very important both in mathematical theory and mathematical applications. The main object of this project is to investigate the ideal Euler fluid with electric external force. Mathematical model we consider is Euler-Poisson equations, the study of such equations have extensive and important applications, and associated fluid problems with our daily life is closely linked in astronomy aviation and others. This project will focus on the Euler-Poisson equations with the following two issues, one is subsonic case and the other is supersonic case, the well-posedness of solutions will be approached. Among them, the subsonic flow problem is corresponding to elliptic system, meanwhile the supersonic problem is corresponding to the hyperbolic-elliptic coupled system. As for the two typical coupled system, related mathematical theory and tools are incomplete. The project will focus on the research in multi-dimensional irrotational fluids and two-dimensional flows with vorticity, under suitable physical boundary conditions. Our aim is to establish the theory of well-posedness for nozzle flows. Such results have significant theoretical and practical value of mathematics, such as the propagation of electrons in submicron semiconductor devices and the biological transport of ions for channel proteins.

守恒律方程（组）解的适定性问题（存在性、唯一性及正则性）在数学理论和数学应用方面都非常重要。本项目主要研究对象是电场外力作用下的理想流体。具体来说我们将要研究管道中欧拉-泊松方程组的适定性。我们计划针对该方程组，研究亚音速和超音速解的适定性问题。其中，亚音速流体对应方程组是椭圆型方程组，超音速流体对应的方程组是双曲-椭圆耦合型方程组。研究这两类耦合型方程组的数学理论和工具还不够完备，我们的研究将围绕高维无旋流体和二维有旋流体展开，结合方程组自身的数学结构，在物理边界条件下，建立适定性理论。此项目研究的结果具有重要的数学理论和应用价值，例如亚微米半导体中电子的传播和蛋白质通道中离子的生物运输。

本项目主要研究欧拉方程组及欧拉-泊松方程组的相关模型的解的适定性问题。这类问题不仅是应用数学理论研究中的重要课题之一，在流体力学、物理学、生物学、工程等领域中也具有广泛的应用价值。受项目资助以来，申请人与合作者研究了高维管道中的亚音速与超音速无旋流体，证明了其解的存在性、唯一性和结构稳定性；研究了一类拟一维管道中的跨音速激波，发现了电场效应对激波解的稳定性的影响，并证明了一定物理条件下激波解的动态稳定性；同时，对于无穷长轴对称管道中的稳态欧拉系统，证明了三维可压缩亚音速流体大旋度解的适定性，以及含非平凡涡流速度的解的适定性，这是国际上首个相关问题的结果；此外，还建立了一类可压缩型生物浅水波方程组解的局部适定性及爆破准则，该结果为研究粘性流体中细胞和细菌在生物信号作用下的运动提供了理论依据。