2D Navier-Stokes方程的渐近吸引子

考虑了2D周期边界条件下Navier-Stokes方程渐近吸引子，即构造了一个有限维解序列，首先证明了该序列不会远离方程的整体吸引子，然后证明了它在长时间后无限趋于方程的整体吸引子，并给出了渐近吸引子的维数估计．     

带弱阻尼Navier-Stokes方程拉回吸引子的收敛性

该文研究了带弱阻尼Navier-Stokes方程解的长时间动力学行为.在外力项及初值的适当假设条件下,利用Galerkin方法证明了弱解的整体适定性及正则性,并根据吸引子基本理论验证了拉回吸引子的存在性及收敛性.     

带弱耗散的两维非自治不可压Navier-Stokes方程拉回吸引子的上半连续性

该文研究了描述流体力学规律的一类带有弱耗散和扰动外力项的两维非自治不可压Navier-Stokes方程拉回吸引子的上半连续性.利用半群(过程族)的分解方法以及弱连续方法,可以得到自治系统全局吸引子和非自治系统拉回吸引子的存在性,进一步地,当ε0收敛到ε=0时候,非自治系统的拉回吸引子A_ε(t)可以连续收敛到自治系统的全局吸引子A.     

三维地磁流方程的整体吸引子

在本文中,我们通过引入一类Ｆｒｅｃｈｅｔ空间,得到了三维有界光滑区域上的地磁流方程的弱解的整体吸引子的存在性。     

阻尼Navier-Stokes方程全局吸引子的正则性（英文）

本文研究阻尼Navier-Stokes方程全局吸引子问题.利用迭代法和线性算子半群的正则性估计,结合经典的全局吸引子理论,证明了阻尼NS方程在H~k空间中存在全局吸引子,并在H~k范数下吸引任意有界集.     

流面上的Navier-Stokes方程及其维数分裂方法

该文首先提出了流面和流层的概念,然后推导出了半测地坐标系下流层内的三维NS (Navier-Stokes)方程,以及流面上的二维NS方程.通过引入流面上的流函数,得到了流函数方程的非线性初边值问题,并讨论了方程解的存在性和唯一性.基于以上讨论,提出了求解三维NS方程的维数分裂方法, 并给出了算例.     

二维不可压缩Navier-Stokes方程的七模类Lorenz方程组的动力学行为及其数值模拟

本文研究了平面不可压缩的Navier-Stokes方程一个七模类Lorenz方程组的混沌行为问题.利用模式截断的方法,获得了一个七模类Lorenz方程组,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.基于分岔图、最大李雅普诺夫指数、庞加莱截面、功率谱揭示了系统混沌行为的普适特征,仿真分析了系统动力学行为的演化过程.     

关于粘性热传导流体可压缩Navier-Stokes方程的最大吸引子的注记

This paper is concerned with the existence of maximal attractors in H^i (i=1,2,4) for the compressible Navier-Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains Ωn in R^n(n=2,3).     

简化Navier-Stokes方程的一般理论

本文研究简化Navier-Stokes方程的一般形式及数学物理背景。所提出的张量型简化方程能够适应一般固壁情况,且具有最大的简化效果。文中运用主次特征法并结合力学背景,阐明了简化方程的影响域和决定域,为相应数值方法提供基本依据。还通过对流扩散过程的细致分析,表明流场中扩散效应具有顺着流动方向的影响域,向流动上游的传播十分有限;Reynolds准则的倒数表征粘性效应向流动上游的传播距离、以及顺流方向粘性效应与垂直流动方向粘性效应的相对强弱;从而说明了较大Reynolds数下完全Navier-Stokes方程向简化Navier-Stokes方程的自然转化,并将简化的概念推广于传热传质。     

On global attractors of the 3D Navier-Stokes equations

In view of the possibility that the 3D Navier-Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all solutions on the weak global attractor are strongly continuous. We also introduce and study a two-parameter family of models for the Navier-Stokes equations, with similar properties and open problems. These models always possess weak global attractors, but on some of them every solution blows up (in a norm stronger than the standard energy one) in finite time.     

热方程的一些端点估计及其在Navier-Stokes方程中的应用——献给陆善镇教授75华诞

本文首先讨论热方程初值问题的解在Hardy、BMO（bounded mean oscillation）和Besov型空间中的估计.然后本文结合Coifmann-Lions-Meyer-Semmes在Hardy空间中的补偿紧性结果,给出Navier-Stokes方程整体弱解的二阶导数的一些端点估计.     

The Stability of Navier-Stokes Equations and the Estimation of Its Attractor Dimension

In this paper, for a class of exterior force term 2s²W^'_{s,s} we analyse the existence of unstable modes of linearized Navier-Stokes Equations (NSE), and associate them with integer points in plane. Furthermore we give the lower boundary dimension estimation of the attractor of NSE. Liu discussed the condition where the exterior force term is W^'_{0,s} in (1, 2), but his method can't be extended to the condition where the exterior force term is W^'_{s_1,s_2} (s_1 ≠ 0, s_2 ≠ 0). So this paper may look as the extention of [1, 2]. The method which we give in this paper has direct application for further study of other properties of NSE (such as Hopf bifurcation). See [3].     

Analytical solutions to the Navier-Stokes equations for non-Newtonian fluid

The pressureless Navier-Stokes equations for non-Newtonian fluid are studied. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper. With the previous results for the analytical blowup solutions of the N-dimensional （N ≥ 2） Navier-Stokes equations, we extend the similar structure to construct an analytical family of solutions for the pressureless Navier-Stokes equations with a normal viscosity term （μ（ρ）｜ u｜^α u）.     

A Five-Dimensional System of Navier-Stokes Equation for a Two-Dimensional Incompressible Fluid on a Torus

A five-mode truncation of Navier-Stokes equation for a two-dimensional incompressible fluid on a torus is studied. Its stationary solutions and stability are presented, the existence of attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through an involved sequence of bifurcations with the changing of Reynolds number, is simulated numerically. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system are revealed.     

Relaxed Euler systems and convergence to Navier-Stokes equations

We consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation both in compressible and incompressible cases. This requires to decompose the second-order derivative terms of the velocity into first-order ones. Usual decompositions lead to approximate systems with tensor variables. We construct approximate systems with vector variables by using Hurwitz-Radon matrices. These systems are written in the form of balance laws and admit strictly convex entropies, so that they are symmetrizable hyperbolic. For smooth solutions, we prove the convergence of the approximate systems to the Navier-Stokes equations in uniform time intervals. Global-in-time convergence is also shown for the initial data near constant equilibrium states of the systems. These convergence results are established not only for the approximate systems with vector variables but also for those with tensor variables.     

The existence of global attractors for 2D Navier-Stokes equations in <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> spaces

In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk（Ω,R2） for any k ≥ 1, which attracts any bounded set of Hk（Ω,R2） in the H^k-norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong＇s conclusion.     

State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations

In this paper, an optimal control problem for the stationary Navier-Stokes equations in the presence of state constraints is investigated. Existence of optimal solutions is proved and first order necessary conditions are derived. The regularity of the adjoint state and the state constraint multiplier is also studied. Lipschitz stability of the optimal control, state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated.     

Qualitative properties of stationary measures for three-dimensional Navier-Stokes equations

The paper is devoted to studying the distribution of stationary solutions for 3D Navier-Stokes equations perturbed by a random force. Under a non-degeneracy assumption, we show that the support of such a distribution coincides with the entire phase space, and its finite-dimensional projections are minorised by a measure possessing an almost surely positive smooth density with respect to the Lebesgue measure. Similar assertions are true for weak solutions of the Cauchy problem with a regular initial function. The results of this paper were announced in the short note [A. Shirikyan, Controllability of three-dimensional Navier-Stokes equations and applications, in: Sémin. Équ. Dériv. Partielles, 2005-2006, École Polytech., Palaiseau, 2006].