A spatial decay estimate for the Navier-Stokes equations

An expression for the spatial decay of an energy functional of solutions of the Navier-Stokes equations is given. The decay is exponential in nature with a decay constant which depends only on the geometry of the flow region.

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Résumé Nous exprimons la diminution par rapport à la distance d'une fonction d'énergie de certaines solutions des équations de Navier-Stokes. L'expression est du type exponentiel décroissant dont le coefficient ne dépend que de la forme de la région d'écoulement.

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This work was supported in part by the Department of the Navy, Naval Sea Systems Command, under Contract No. N00024-78-C-5384.     

Étude de l'écoulement laminaire de deux liquides non miscibles entre deux disques coaxiaux tournant en sens contraires

Resumé On présente des résultats théoriques relatifs à l'écoulement laminaire permanent de deux liquides non miscibles entre disques tournant en sens contraires. La solution cherchée est une solution exacte des équations de Navier-Stokes appartenant à la classe des solutions affines représentant les écoulements entre disques, avec symétrie de révolution. Cette solution est calculée analytiquement de manière approchée pour les valeurs faibles du nombre de Reynolds de rotation, et numériquement pour les valeurs quelconques de ce paramètre. Les résultats sont analysés en fonction des valeurs relatives des masses volumiques et des viscosités des deux fluides.

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This paper is concerned with a theoretical study of the stationary laminar flow of two immiscible liquids between two disks rotating in opposite senses. The solution is an exact solution of the complete Navier-Stokes equations belonging to the class of similar solutions of flows between rotating coaxial disks. An approximate analytical form of this solution is given for low values of the rotation Reynolds number and numerical results have been performed for higher values of this parameter. Detailed results are discussed for relative values of fluids densities and viscosities.

     

Exact solutions of the unsteady two-dimensional Navier-Stokes equations

This paper studies the two-dimensional incompressible viscous flow in which the local vorticity is proportional to the stream function perturbed by a uniform stream. It was known by Taylor and Kovasznay that the Navier-Stokes equations for flow of this kind become linear. From the general solution to the linear equations for steady flow, we show that there exist only two types of steady flow of this kind: Kovasznay downstream flow of a two-dimensional grid and Lin and Tobak reversed flow about a flat plate with suction. In the unsteady flow case, new classes of exact analytical solutions are found which include Taylor vortex array solution as a special case. It is shown that these unsteady flows are, as viewed from a frame of reference moving with the undisturbed uniform stream, pseudo-steady in the sense that the flow pattern is steady but the magnitude of motion decays, or grows, exponentially in time. All these solutions are valid for any Reynolds number.

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Résumé Dans ce travail nous étudions l'écoulement plan d'un fluide visqueux incompressible dans lequel la rotation locale est proportioneile à la fonction de courant perturbée par un courant uniforme. Conformément aux travaux de Taylor et Kovasznay les équations de Navier-Stokes pour cet écoulement deviennent linéaires. Par conséquent nous utilisons la solution générale pour démontrer que seulement deux catégories d'écoulement stationnaire peuvent exister: l'écoulement de Kovasznay en aval d'une grille plane, et l'écoulement inversé de Lin et Tobak pour une plaque plane avec aspiration. Nous étudions aussi l'écoulement non stationnaire et nous découvrons des classes nouvelles de solutions exactes qui contiennent, en particulier, le réseau de tourbillons de Taylor. Enfin nous démontrons que ces écoulements sont pseudo-stationnaires dans un système de coordonnées en mouvement avec le courant uniforme non perturbé; ce qui signifie que l'amplitude de l'écoulement stationnaire croit ou décroit exponentiellment dans le temps. Toutes ces solutions sont valides pour tous les nombres de Reynolds.

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On leave from University of Waterloo, Ontario, Canada.     

Eccentric rotating flows: Exact unsteady solutions of the Navier-Stokes equations

We present some exact solutions of the Navier-Stokes equations which describe the development of eccentric flows in a rotating fluid. In particular, it is seen how an eccentric solid body rotation behaviour can be developed.

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Resumé On décrit le développement de l'écoulement excentrique dans un liquide tournant quand il y a des axes différents. Des solutions exactes des équations de Navier et Stokes s'offrirent; une solution particulière représente l'écoulement excentrique d'une masse solide.

     

Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise

In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus perturbed by a Lévy process. The existence of invariant measure of the solutions are proved also. This work was supported by National Basic Research Program of China (Grant No. 2006CB8059000), Science Fund for Creative Research Groups (Grant No. 10721101), National Natural Science Foundation of China (Grant Nos. 10671197, 10671168), Science Foundation of Jiangsu Province (Grant Nos. BK2006032, 06-A-038, 07-333) and Key Lab of Random Complex Structures and Data Science, Chinese Academy of Sciences     

Large deviations for stochastic PDE with Lévy noise

We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Lévy noise. We use general large deviations theorems of Varadhan and Bryc coupled with the techniques of Feng and Kurtz (2006) [15], viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton-Jacobi-Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Lévy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.     

Homotopy Analysis Method for Solving(2+1)-dimensional Navier-Stokes Equations with Perturbation Terms

In this paper Homotopy Analysis Method (HAM) is implemented for obtaining approximate solutions of (2+1)-dimensional Navier-Stokes equations with perturbation terms. The initial approximations are obtained using linear systems of the Navier-Stokes equations;by the iterations formula of HAM,the first approxima-tion solutions and the second approximation solutions are successively obtained and Homotopy Perturbation Method(HPM)is also used to solve these equations;finally, approximate solutions by HAM of (2+1)-dimensional Navier-Stokes equations with-out perturbation terms and with perturbation terms are compared. Because of the freedom of choice the auxiliary parameter of HAM,the results demonstrate that the rapid convergence and the high accuracy of the HAM in solving Navier-Stokes equa-tions;due to the effects of perturbation terms,the 3rd-order approximation solutions by HAM and HPM have great fluctuation.     

On the time inhomogeneous skew Brownian motion

This paper is devoted to the construction of a solution for the “Inhomogeneous skew Brownian motion” equation, which first appeared in a seminal paper by Sophie Weinryb, and recently, studied by Étoré and Martinez. Our method is based on the use of the Balayage formula. At the end of this paper we study a limit theorem of solutions.     

On the rotating Navier-Stokes equations with mixed boundary conditions

The stationary and nonstationary rotating Navier-Stokes equations with mixed boundary conditions are investigated in this paper. The existence and uniqueness of the solutions are obtained by the Galerkin approximation method. Next, θ-scheme of operator splitting algorithm is applied to rotating Navier-Stokes equations and two subproblems are derived. Finally, the computational algorithms for these subproblems are provided.     

Estimates of suitable weak solutions to the Navier-Stokes equations in critical Morrey spaces

We prove some estimates for suitable weak solutions to the nonstationary three-dimensional Navier-Stokes equations under the assumption that certain invariant functionals of the velocity field are bounded. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 199–210.     

Existence of global weak solutions to one-component vlasov-poisson and fokker-planck-poisson systems in one space dimension with measures as initial data

We consider Cauchy problems for the 1-D one component Vlasov-Poisson and Fokker-Planck-Poisson equations with the initial electron density being in the natural space of arbitrary non-negative finite measures. In particular, the initial density can be a Dirac measure concentrated on a curve, which we refer to as “electron sheet” initial data. These problems resemble both structurally and functional analytically Cauchy problems for the 2-D Euler and Navier-Stokes equations (in vorticity formulation) with vortex sheet initial data. Here, we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved. We give a natural definition of the product, establish its weak stability, and existence of weak solutions follows. Our concept of weak solutions through the newly defined product is justified since solutions to the Fokker-Planck-Poisson equation, the analogue of Navier-Stokes equation, are shown to converge to weak solutions of the Vlasov-Poisson equation as the Fokker-Planck term vanishes. The main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations. This is needed because the problem studied here is beyond the range of applicability of the “velocity averaging” compactness methods of DiPerna-Lions. © 1994 John Wiley & Sons, Inc.     

Combined finite element and spectral approximation of the Navier-Stokes equations

Summary We present a method for the numerical approximation of Navier-Stokes equations with one direction of periodicity. In this direction a Fourier pseudospectral method is used, in the two others a standard F.E.M. is applied. We prove optimal rate of convergence where the two parameters of discretization intervene independently.

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Approximation des équations de Navier-Stokes par une méthode éléments finis-spectrale Fourier

Resumé On présente une méthode d'approximation numérique des équations de Navier-Stokes possédant une direction de périodicité. Dans cette direction une méthode pseudospectrale basée sur des développements en série de Fourier est utilisée, dans les deux autres on applique une méthode d'éléments finis standard. On montre que la convergence est optimale et que les deux paramètres de discrétisation peuvent être choisis de façon indépendante.

     

Theoretical study of a Bénard-Marangoni problem

In this paper we prove the existence of strong solutions for the stationary Bénard-Marangoni problem in a finite domain flat on the top, bifurcating from the basic heat conductive state. The Bénard-Marangoni problem is a physical phenomenon of thermal convection in which the effects of buoyancy and surface tension are taken into account. This problem is modelled with a system of partial differential equations of the type Navier-Stokes and heat equation. The boundary conditions include crossed boundary conditions involving tangential derivatives of the temperature and normal derivatives of the velocity field. To define tangential derivatives at the boundary, intended in the trace sense, it is necessary order two derivatives in the interior of the domain and thus the boundary term contains as high derivatives as the interior term. We overcome this difficulty by considering the weak formulation, and transforming the boundary integral into an equivalent integral defined in the whole domain. This allows us to reformulate the weak problem with a temperature having only order one weak derivatives. Concerning regularity results, we obtain strong solutions for the stationary Bénard-Marangoni problem.     

Remark on compressible Navier-Stokes equations with density-dependent viscosity and discontinuous initial data

In this paper, we study the free boundary problem for 1D compressible Navier-Stokes equations with density-dependent viscosity. We focus on the case where the viscosity coefficient vanishes on vacuum. We prove the global existence and uniqueness for discontinuous solutions to the Navier-Stokes equations when the initial density is a bounded variation function, and give a decay result for the density as t→+∞.     

Asymptotic integration of Navier-Stokes equations with potential forces. II. An explicit Poincaré-Dulac normal form

We study the incompressible Navier-Stokes equations with potential body forces on the three-dimensional torus. We show that the normalization introduced in the paper [C. Foias, J.-C. Saut, Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987) 1-47], produces a Poincaré-Dulac normal form which is obtained by an explicit change of variable. This change is the formal power series expansion of the inverse of the normalization map. Each homogeneous term of a finite degree in the series is proved to be well-defined in appropriate Sobolev spaces and is estimated recursively by using a family of homogeneous gauges which is suitable for estimating homogeneous polynomials in infinite dimensional spaces.     

Incompressible limits of the Navier-Stokes equations for all time

We study the asymptotic behaviors of the regular solutions to the compressible Navier-Stokes equations for “well-prepared” initial data for all time as the Mach number tends to zero, by deriving a differential inequality with certain decay property. The estimates obtained in this paper are uniform both in time and Mach number.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.     

Étude de l'établissement d'un jet liquide laminaire émergeant d'une conduite cylindrique verticale semi-infinie et soumis à l'influence de la gravité

Résumé On étudie l'écoulement libre d'un jet liquide laminaire axisymétrique émergeant d'une conduite cylindrique verticale semi-infinie, à grands nombres de Reynolds et grands nombres de Froude. Dans de telles conditions, si l'on exclut le cas des tubes capillaires, le nombre de Weber de l'écoulement est suffisamment grand pour que les effets de tension superficielle s'avèrent négligeables, ainsi que le confirme cette étude. Les équations sont simplifiées par une analyse de type couche limite. L'utilisation de la méthode des développements asymptotiques raccordés conduit à des solutions approchées semianalytiques faisant apparaitre les paramètres de similitude de l'écoulement. Ces solutions prévoient la distribution des vitesses et la forme du jet sous l'influence des forces de pesanteur et de viscosité. Des résultats expérimentaux obtenus par photographie du jet et visualisation holographique du champ des vitesses confirment les prévisions théoriques.

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This paper is concerned with the laminar flow of a free liquid jet emerging from a semi infinite vertical cylindrical pipe at high Reynolds and Froude numbers. In that case, excepting the capillary tubes, the Weber number is sufficiently high so that surface tension effects may be neglected. The general equations are simplified by a boundary layer analysis and solved by using the matched asymptotic expansion's method. Approximate semi analytical solutions are presented showing parameters of similarity. These solutions predict the velocity distribution and the jet shape under the influence of body and viscous forces. Experimental results obtained by jet photography and holographic visualization of the velocity field confirm the accuracy of the calculations.

     

Large deviation principles for 2-D stochastic Navier-Stokes equations driven by Lévy processes

In this paper, we establish a large deviation principle for the two-dimensional stochastic Navier-Stokes equations driven by Lévy processes, which involves the study of the Lévy noise and the investigation of the effect of the highly nonlinear, unbounded drifts.     

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）.