流体力学方程组的适定性问题与极限问题

项目名称： 流体力学方程组的适定性问题与极限问题

项目编号： No.11471334

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 欧耀彬

作者单位： 中国人民大学

项目金额： 60万元

中文摘要： 流体力学方程组的理论研究，一直以来都是偏微分方程理论的最重要课题之一。本项目将研究一些可压缩流体方程组的适定性问题和渐近极限问题, 后者包括大时间渐近极限问题和流体力学极限问题。首先，本项目要研究可压缩流体方程组（例如Navier-Stokes方程组、粘弹性流体方程组）的低马赫数极限和相关的流体力学极限问题，尤其是在物理固壁边界条件下的情形。在这些极限过程中，方程组的解会表现出奇异性，因而在数学物理上是很有意义且很有挑战性的课题。其次，我们将研究可压缩流体方程组（如浅水波方程组、Navier-Stokes方程组）的自由边值问题光滑解或强解的局部和整体存在性及其大时间渐近行为。此类问题目前关于光滑解或强解的结果较少，因此要研究这类问题必须利用、发展新的思想和方法。

中文关键词： 可压缩流体；适定性；渐近极限；自由边值问题；马赫数

英文摘要： The research on hydrodynamic equations is always among the most important topics on the theory of partial differential equations. In this project, we study well-posedness problems and asymptotic limits, including large-time asymptotics and hydrodynamic limits, for compressible hydrodynamic equations. First, we study the low Mach number limit and related hydrodynamic limits of compressible hydrodynamic equations,e.g., Navier-Stokes equations and viscoelastic hydrodynamic equations. In particular, we are interested in the situations with solid boundary conditions. In the limit process, the solutions usually present singularities. Thus these problems are very interesting and challenging in Mathematical Physics. Next, we study the global existence and large-time stability for smooth solutions or strong solutions to free boundary problems for compressible hydrodynamic equations, e.g., shallow water equations and Navier-Stokes equations. To our best knowledge, there are few results on the well-posedness for smooth solutions or strong solutions of these kinds of problems. Thus we should utilize and develop new ideas and techniques to investigate these topics.

英文关键词： compressible fluids;well-posedness;asymptotic limits;free boundary problem;Mach number