黏性系数依赖密度型可压缩Navier-Stokes 方程的零耗散极限问题

本文考虑黏性系数依赖密度的可压缩Navier-Stokes 方程解的零耗散极限问题. 假定Euler 方程的稀疏波解一端被真空状态连接, 我们证明Navier-Stokes 方程存在一列(依赖黏性的) 整体解, 且随着粘性的消失, 此整体解逐渐稳定于Euler 方程对应的稀疏波解和真空状态; 并且得到了一致衰减率估计. 此结果推广了常黏性系数的情形.     

非定常线性化Navier-Stokes方程的非协调流线扩散有限元法分析

对非定常线性化Navier-Stokes方程提出了非协调流线扩散有限元方法．用向后Euler格式离散时间,用流线扩散法处理扩散项带来的非稳定性．速度采用不连续的分片线性逼近,压力采用分片常数逼近．得到了离散解的存在唯一性以及在一定范数意义下离散解的稳定性和误差估计．     

Navier-Stokes方程的奇异性对大气运动方程组的影响

综述了大气运动基本方程组在光滑函数类中的稳定性和Navier-Stokes方程的不稳定性的若干结论．在此基础上,以大气运动方程组的Boussinesq近似为例,阐述了Navier-Stokes方程的不稳定性导致的大气运动基本方程组的某些简化模式的不稳定性,从而得到在简化基本方程过程中应该遵守的一个原则,以保证简化方程的稳定性．     

不可压流体的边界层问题

研究三维有界区域在边界上有流动的不可压流体的边界层问题,导出了Navier-Stokes方程区域内部的近似方程(Euler方程和线性化的Euler方程)和边界附近近似的方程(零阶边界层方程与一阶边界层方程),证明了这种近似的合理性.     

从Euler K法正方程推导K法正方程

目的是在液体分子的分布符合局部Gibbs分布（正侧系综）的条件下,从Euler K泛函方程推导出描述液体平均密度,平均速度和平均能量演化的方程,作者从K的泛函形式出发,考察了Euler K泛函方程的一些特殊情形。通过这些特殊情形得到了所需的平均速度方程。平均密度方程和平均能量方程。     

Navier-Stokes方程的一类粘性分离算法

本文提出了一类求解Navier-Stokes方程的粘性分离算法,在每一时间层内,将原方程分解为线性 Euler方程和非定常Stokes方程,证明了空间连续格式和完全离散格式的收敛性,并给出了最优阶误差估计。     

Navier-Stokes方程稳定性研究(Ⅱ)

本文对Navier-Stokes方程与热传导方程的性质进行了比较。法国数学家、偏微分方程权威J.Leray教授在其对Navier-Stokes方程的研究中,曾由热传导方程出发而求得Navier-Stokes方程某种初(边)值问题的适定性结果[2].巴黎十一大学的R.Temam等专家、教授也曾多次提出过将两类方程类比的疑问。本文试将其中根本不同点做了叙述和例证。     

隐式Euler法关于Volterra延迟积分方程的数值稳定性

本文涉及隐式Euler法应用于非线性Volterra型延迟积分方程的稳定性,其探讨了基于非经典Lipschitz条件,其方法的整体与渐近稳定性结果被获得。     

二维Euler发展方程的一个超积分谱方法逼近

本文介绍一种解二维Euler发展方程的Legendre-Collocation谱方法，并引进轻微的超数值积分技巧，证明了这种逼近的稳定性和收敛性     

平面不可压缩的NAVIER-STOKES方程新五模类LORENZ方程组的混沌行为

本文研究了平面正方形区域上不可压缩的Navier-Stokes方程五模类Lorenz方程组的混沌行为问题.利用傅立叶展开方法对Navier-Stokes方程进行模式截断,获得了新五模类Lorenz方程组,给出了该方程组定常解及其稳定性的讨论,证明了该方程组吸引子的存在性,并对其全局稳定性进行了分析和讨论.     

Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system

In this paper, we study the vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen-Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier-Stokes and Euler equations, and assuming that the energy dissipation for Navier-Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier-Stokes/Allen-Cahn system converge to that of the Euler/Allen-Cahn system in a proper small time interval. In addition, for strong solutions of the Navier-Stokes/Allen-Cahn system in 2D, the convergence rate is cν1/2.     

Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model

We discuss the inviscid limits for the randomly forced 2D Navier-Stokes equation (NSE) and the damped/driven KdV equation. The former describes the space-periodic 2D turbulence in terms of a special class of solutions for the free Euler equation, and we view the latter as its model. We review and revise recent results on the inviscid limit for the perturbed KdV and use them to suggest a setup which could be used to make a next step in the study of the inviscid limit of 2D NSE. The proposed approach is based on an ergodic hypothesis for the flow of the 2D Euler equation on iso-integral surfaces. It invokes a Whitham equation for the 2D Navier-Stokes equation, written in terms of the ergodic measures.     

Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model

We discuss the inviscid limits for the randomly forced 2D Navier-Stokes equation (NSE) and the damped/driven KdV equation. The former describes the space-periodic 2D turbulence in terms of a special class of solutions for the free Euler equation, and we view the latter as its model. We review and revise recent results on the inviscid limit for the perturbed KdV and use them to suggest a setup which could be used to make a next step in the study of the inviscid limit of 2D NSE. The proposed approach is based on an ergodic hypothesis for the flow of the 2D Euler equation on iso-integral surfaces. It invokes a Whitham equation for the 2D Navier-Stokes equation, written in terms of the ergodic measures. Dedicated to Vladimir Igorevich Arnold on his 70th birthday     

Effective velocity in compressible Navier-Stokes equations with third-order derivatives

A formulation of certain barotropic compressible Navier-Stokes equations with third-order derivatives as a viscous Euler system is proposed by using an effective velocity variable. The equations model, for instance, viscous Korteweg or quantum Navier-Stokes flows. The formulation in the new variable allows for the derivation of an entropy identity, which is known as the BD (Bresch-Desjardins) entropy equation. As a consequence of this estimate, a new global-in-time existence result for the one-dimensional quantum Navier-Stokes equations with strictly positive particle densities is proved.     

Darcy渗流定律的微观界定及其应用

将Boltzmann微观方程用Chapman—Enskog展开，结合BGK近似理论得到Navier-Stokes方程，通过在多孔介质的某种表征体元上的不同的方式平均Navier-Stokes方程后得出Darcy渗流定律的一种表示方法，并用实例证明了方法的可靠性。     

Numerical Solution of Several Models of Internal Transonic Flow

The paper deals with numerical solution of internal flow problems. It mentions a long tradition of mathematical modeling of internal flow, especially transonic flow at our department. Several models of flow based on potential equation, Euler equations, Navier-Stokes and Reynolds averaged Navier-Stokes equations with proper closure are considered. Some mathematical and numerical properties of the model are mentioned and numerical results achieved by in-house developed methods are presented.     

ON THE VISCOSITY SPLITT NG METHOD FOR INITIAL BOUNDARY VALUE PROBLEMS OF THE NAVIAR-STOKES EQUATIONS

The viscosily splitting method for the Navier-Stokes equations on two dimensional multi-connected domains is considered. The equation is split into an Euler equation and a non-stationary Stekes equation within each time step. The author proves the convergence theorem as he has done for the problem on simply connected domains, and the rate of convergence is improved from loss than 1/4 to 1.     

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 252–264, May, 2008.     

A variational principle for the Navier-Stokes equation

Motivated from Arnold's variational characterization of the Euler equation in terms of geodesic families of diffeomorphisms, a variational principle for the motion of incompressible viscous fluids is presented. A volume preserving diffusion process with drift velocity field subject to the Navier-Stokes equation is shown to extremize the energy functional of the fluid under a certain class of stochastic variations.     

Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations

The main purpose of this paper is to study the asymptotic equivalence of the Boltzmann equation for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t=0) as ?→0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. In this setting, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat conductivity (depending on the viscosity) is also obtained.