Ergodicity of the 2-D Navier-Stokes equation under random perturbations

A 2-dimensional Navier-Stokes equation perturbed by a sufficiently distributed white noise is considered. Existence of invariant measures is known from previous works. The aim is to prove uniqueness of the invariant measures, strong law of large numbers, and convergence to equilibrium.     

Global Well-Posedness for a Smoluchowski Equation Coupled with Navier-Stokes Equations in 2D

We prove global existence for a nonlinear Smoluchowski equation (a nonlinear Fokker-Planck equation) coupled with Navier-Stokes equations in 2d. The proof uses a deteriorating regularity estimate in the spirit of [5] (see also [1]).     

On Distribution of Energy and Vorticity for Solutions of 2D Navier-Stokes Equation with Small Viscosity

We study distributions of some functionals of space-periodic solutions for the randomly perturbed 2D Navier-Stokes equation, and of their limits when the viscosity goes to zero. The results obtained give explicit information on distribution of the velocity field of space-periodic turbulent 2D flows.     

On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NS _ν) with initial data in the scaling invariant Besov space, here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANS _ν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, and Then with initial data in the scaling invariant space we prove the global wellposedness for (ANS _ν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANS _ν) with high oscillatory initial data (1.2).     

Self-Similar Solutions of Three-Dimensional Navier-Stokes Equation

In this article we will present pure three dimensional analytic solutions for the Navier-Stokes and the continuity equations in Cartesian coordinates.
The key idea is the three-dimensional generalization of the well-known self-similar Ansatz of Barenblatt. A geometrical interpretation of the Ansatz is given also. The results are the Kummer functions or strongly related. Our final formula is compared with other results obtained from group theoretical approaches.     

Pipe Flow and Wall Turbulence Using a Modified Navier-Stokes Equation

We use a derived incompressible modified Navier-Stokes equation to model pipe flow and wall turbulence.We reproduce the observed flattened paraboloid velocity profiles of turbulence that cannot be obtained directly using standard incompressible Navier-Stokes equation.The solutions found are in harmony with multi-valued velocity fields as a definition of turbulence.Repeating the procedure for the flow of turbulent fluid between two parallel flat plates we find similar flattened velocity profiles.We extend the analysis to the turbulent flow along a single wall and compare the results with experimental data and the established controversial von Karman logarithmic law of the wall.     

A Variational Formulation for the Navier-Stokes Equation

In this paper we prove a new variational principle for the Navier-Stokes equation which asserts that its solutions are critical points of a stochastic control problem in the group of area-preserving diffeomorphisms. This principle is a natural extension of the results by Arnold, Ebin, and Marsden concerning the Euler equation.Supported in part by FCT/POCTI/FEDER     

On a potential-velocity formulation of Navier-Stokes equations

Computational methods in continuum mechanics, especially those encompassing fluid dynamics, have emerged as an essential investigative tool in nearly every field of technology. Despite being underpinned by a well-developed mathematical theory and the existence of readily available commercial software codes, computing solutions to the governing equations of fluid motion remains challenging: in essence due to the non-linearity involved. Additionally, in the case of free surface film flows the dynamic boundary condition at the free surface complicates the mathematical treatment notably. Recently, by introduction of an auxiliary potential field, a first integral of the two-dimensional Navier-Stokes equations has been constructed leading to a set of equations, the differential order of which is lower than that of the original Navier-Stokes equations. In this paper a physical interpretation is provided for the new potential, making use of the close relationship between plane Stokes flow and plane linear elasticity. Moreover, it is shown that by application of this alternative approach to free surface flows the dynamic boundary condition is reduced to a standard Dirichlet-Neumann form, which allows for an elegant numerical treatment. A least squares finite element method is applied to the problem of gravity driven film flow over corrugated substrates in order to demonstrate the capabilities of the new approach. Encapsulating non-Newtonian behaviour and extension to three-dimensional problems is discussed briefly.     

Some Similarity Reduction Solutions to Two-Dimensional Incompressible Navier-Stokes Equation

We investigate the symmetry reduction for the two-dimensional incompressible Navier Stokes equation in conventional stream function form through Lie symmetry method and construct some similarity reduction solutions. Two special cases in [D.K. Ludlow, P.A. Clarkson, and A.P. Bassom, Stud. Appl. Math. 103 （1999） 183] and a theorem in [S.Y. Lou, M. Jia, X.Y. Tang, and F. Huang, Phys. Rev. E 75 （2007） 056318] are retrieved.     

隐式格式求解拟压缩性非定常不可压Navier-Stokes方程

采用Rogers发展的双时间步拟压缩方法,数值求解不可压非定常问题.数值通量分别采用三阶精度Roe格式和二阶精度Harten-Yee的TVD格式离散.为了加快收敛,提高求解效率,试验了几种隐式格式(ADI-LU,LGS,LU-SGS).针对经典的低雷诺数(Re=200)圆柱绕流问题,比较了不同隐式方法的计算结果和求解效率,以及两种数值离散格式计算结果的异同.最后采用Roe格式数值求解了两种典型的低速非定常流动问题:绕转动圆柱(ω=1)低雷诺数流动;NACA0015翼型等速拉起数值模拟.     

Some Similarity Reduction Solutions to Two-Dimensional Incompressible Navier-Stokes Equation

We investigate the symmetry reduction for the two-dimensional incompressible Navier-Stokes equation in conventional stream function form through Lie symmetry method and construct some similarity reduction solutions. Two special cases in [D.K. Ludlow, P.A. Clarkson, and A.P. Bassom, Stud. Appl. Math. 103 (1999) 183] and a theorem in [S.Y. Lou, M. Jia, X.Y. Tang, and F. Huang, Phys. Rev. E 75 (2007) 056318] are retrieved.     

On a new derivation of the Navier-Stokes equation

Given a net of finite-dimensional real Lie algebras contracting into a Lie algebra, a representation of is constructed explicitly as limit of a net ( _l ) of representations, each _l being a representation of on a complex Hilbert space _l . Conditions are imposed on the net ( _l ) implying that the carrier space of contain a-stable set of vectors which are analytic for all, where is a basis of. As a corollary, the corresponding result for contractions of representations of simply connected finite-dimensional real Lie groups is derived.Supported by the Swiss National Science Foundation     

Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space.¶ II. Construction of the Navier-Stokes Solution

This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem. Received: 5 September 1996 / Accepted: 14 July 1997     

No Existence and Smoothness of Solution of the Navier-Stokes Equation

The Navier-Stokes equation can be written in a form of Poisson equation. For laminar flow in a channel (plane Poiseuille flow), the Navier-Stokes equation has a non-zero source term (∇²u(x, y, z) = F_x (x, y, z, t) and a non-zero solution within the domain. For transitional flow, the velocity profile is distorted, and an inflection point or kink appears on the velocity profile, at a sufficiently high Reynolds number and large disturbance. In the vicinity of the inflection point or kink on the distorted velocity profile, we can always find a point where ∇²u(x, y, z) = 0. At this point, the Poisson equation is singular, due to the zero source term, and has no solution at this point due to singularity. It is concluded that there exists no smooth orphysically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain due to singularity.     

Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called Oseens vortex. This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we give precise estimates on the rate of convergence toward the vortex.Acknowledgement The first author is indebted to J. Dolbeault and, especially, to C. Villani for suggesting the beautiful idea of using the Boltzmann entropy functional in the context of the two-dimensional Navier-Stokes equation. The research of C.E.W. is supported in part by the NSF under grant DMS-0103915.     

二维球坐标系中子输运方程的一种并行SN算法

针对二维球坐标系下中子输运方程的SN算法, 提出基于(单元, 方向)二元组的有向图模型, 在已有的基于有向图的并行流水线算法基础上, 设计粒度可控多级并行SN算法。其中, 采用区域分解和并行流水线相结合的方式挖掘空间-角度方向的并行度, 提出能群流水并行方法, 并通过设置合适的流水线粒度来平衡有向图调度、通信和空闲等待开销。实验结果表明: 该算法可以有效地求解二维球坐标系下的中子输运方程。在某国产并行机1920核上, 对于96万网格、60个方向、24能群、数十亿自由度的典型中子输运问题, 获得了71%的并行效率。