两类非线性双曲-抛物耦合方程的粘性消失极限问题

项目名称： 两类非线性双曲-抛物耦合方程的粘性消失极限问题

项目编号： No.11301172

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 张映辉

作者单位： 湖南理工学院

项目金额： 22万元

中文摘要： 力学中有无数有趣而且有意义的非线性双曲-抛物耦合偏微分方程的问题值得研究和探讨。本项目主要研究两个重要的例子，其一是在航空航天、地质力学和图像处理等诸多方面都有用武之地的流体动力学Navier-Stokes 方程组（双曲-抛物耦合方程组），其二是在研究身体细胞、细菌及其他单细胞、多细胞生物依据环境中某些化学物质而趋向的运动过程等问题中有重要应用的生物力学Chemotaxis方程组（双曲-抛物耦合方程组）。关于上述两类方程初值问题的适定性和粘性消失极限的研究，已有的工作大部分集中于"光滑初值"问题上，至于"非光滑初值"问题，至今还有很多重要的问题未得到解决。为此，我们有必要发展新的思想、技巧和方法，深入地研究其数学结构和特性，为推动其数学理论的基础和应用研究作出一些贡献。本项目将主要研究上述两类方程 Riemann 问题的适定性和粘性消失极限。

中文关键词： Navier-Stokes方程；生物力学Chemotaxis方程；适定性；粘性消失极限；

英文摘要： There are countless interesting and meaningful problems of nonlinear hyperbolic-parabolic coupled partial differential equations in Mechanics which are worth researching and discussing. This project mainly studies two important examples, one of which is the Navier-Stokes equations (hyperbolic-parabolic coupled equations) in fluid dynamics which is extensively used in many aspects such as Aeronautics and Astronautics, Geological mechanics,Image processing and etc., and the second is the Chemotaxis equations (hyperbolic-parabolic coupled equations) in biomechanics which has important applications in studying tendency of movement based on certain chemicals in the environment of the body's cells, bacteria and other single-celled and multi-cellular organisms, and other problems. For the studies on the well-posedness and vanishing viscosity limit of the initial value problem to the above two types of equations, most of the previous works were focused on the "smooth initial value" problem. So far, there are still many important issues on "non-smooth initial value" problem unresolved. To this end, we need to develop new ideas, techniques and methods to study deeply the mathematical structure and characteristics so as to make some contributions to promote basic and applied research of the mathematical theory. This projec

英文关键词： Navier-Stokes equations；biomechanical Chemotaxis equations；well-posedness；vanishing viscosity limit；