管道中欧拉-泊松流体的适定性问题

项目名称： 管道中欧拉-泊松流体的适定性问题

项目编号： No.11501077

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 段犇

作者单位： 大连理工大学

项目金额： 18万元

中文摘要： 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。

中文关键词： Euler-Poisson方程组；适定性；管道流体；超音速流体；亚音速流体

英文摘要： 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.

英文关键词： Euler-Poisson equations;well-posedness;nozzle flow;supersonic flow;subsonic flow