两类非线性双曲-抛物耦合方程的粘性消失极限问题

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 project will mainly study the well-posedness and vanishing viscosity limit of Riemann problem to the above two types of equations.

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。

Navier-Stokes方程和生物力学Chemotaxis方程的适定性和粘性消失极限问题一直是偏微分方程领域同行研究的热点。本项目的研究成果主要包含三个方面的内容: 1. 非等熵 Navier-Stokes 方程具有激波初值的适定性和粘性消失极限问题。项目得到了非等熵 Navier-Stokes 方程激波初值问题解的整体存在性、正则性、跳跃不连续关于粘性的衰减性质及波之间的干扰估计，最后证明了当粘性消失时，Navier-Stokes 方程的解一致收敛到相应的 Euler 方程的 Riemann 解； 2. 两相Navier-Stokes方程初值问题的适定性和大时间行为。项目得到了三维粘性和无粘两相Navier-Stokes方程初值问题解的整体适定性和最优衰减率。3. 生物力学Chemotaxis方程的适定性和大时间行为。项目得到了生物力学Chemotaxis方程初值问题(初边值问题)解的全局存在性和最优衰减率(指数收敛率)。