两类流体动力学方程解的一些渐近性质研究

Nonlinear damped Navier-Stokes (N-S) equations and Brinkman-Forchheimer (B-F) equations are two important fluid dynamics equations.In view of the three-dimensional case, research findings on the long-time behavior of solutions of these two kinds of equations are very less. The current proposed project will provide detailed theoretical analysis on the asymptotic properties of solutions of these two kinds of equations.Firstly, using Galerkin approximation method, we will study the existence and uniqueness of strong solution of three-dimensional Brinkman-Forchheimer equation. Secondly, based on the results of existence and uniqueness of strong solution, using infinite dimensional dynamical system theory, we will discuss the existence of various kinds of attractor of these two kinds of fluid equations.By the mehtod of truncation and uniform estimates on tails of solution, or Palay-Littlewood decomposition and solution operator decomposition method,we will prove the asymptotic compactness of solution operator in the unbounded domain satisfying Poincare's inequality.Finally, by Fourier decomposition method, we will explore the time decay and asymptotic stability of solutions of three-dimensional Brinkman-Forchheimer equation.The research not only possesses high academic significance for the deep exploration of basic law and physical mechanism for the motion of the two kinds of fluid, but also has significant theorectical guidance value for studing other aspects of these two kinds of fluid.

非线性阻尼Navier-Stokes(N-S)方程和Brinkman-Forchheimer(B-F)方程是两类重要的流体力学方程.鉴于三维情形这两类方程解的长时间行为研究成果较少,本项目拟从无穷维动力系统角度出发对这两类方程解的渐近性质进行深入探讨.首先采用Galerkin逼近方法对三维B-F方程强解的存在唯一性进行讨论.其次,基于强解存在唯一性结论,运用无穷维动力系统理论对这两类流体方程强解各种吸引子的存在性进行研究.将采用截断方法与尾估计方法或Palay-Littlewood分解与解算子分解方法对满足Poincare不等式的无界区域中解算子的渐近紧性进行验证.最后,通过傅里叶分解方法对三维B-F方程强解的时间衰减性和初值在大扰动情形下系统的渐近稳定性进行探讨.本项目的研究不仅对深入探索这两类流体运动的基本规律和物理机制具有良好的学术意义,同时对其它方面的研究具有一定的理论指导价值.

本项目主要研究了三维非线性阻尼Navier-Stokes方程和三维Brinkman-Forchheimer方程两类流体力学方程解的长时间行为，从无穷维动力系统角度出发对这两类流体方程解的一些渐近性质进行了深入地探讨。针对三维Brinkman-Forchheimer方程，首先运用Faedo-Galerkin逼近方法讨论了三维全空间上Brinkman-Forchheimer方程弱解及强解的存在性和强解唯一的条件；在此基础上探讨了三维有界区域上Brinkman-Forchheimer方程强解的各种吸引子的存在性；研究了三维有界区域中具有奇异震荡外力项的Brinkman-Forchheimer方程强解的一致吸引子的存在性、一致有界性和收敛性；分析了三维全空间上Brinkman-Forchheimer方程弱解及强解关于时间的衰减性和初值在大扰动情形下系统的渐近稳定性。针对三维非线性阻尼Navier-Stokes方程，首先运用无穷维动力系统理论深入地探讨了该方程强解的各种吸引子的存在性；研究了三维有界区域中具有奇异震荡外力项的三维非线性阻尼Navier-Stokes强解一致吸引子的一致有界性和收敛性。另外，还讨论了另外一种流体方程—二维Newton-Boussinesq方程解的吸引子的存在性，分别在二维有界区域和二维带形区域中得到了Newton-Boussinesq方程在较高的正则性空间上的全局吸引子的存在性。本项目的研究不仅对深入探索这两类流体运动的基本规律和物理机制具有良 好的学术意义,同时对其它方面的研究具有一定的理论指导价值.