Analytic solutions of the Navier-Stokes equations

We consider the time dependent incompressible Navier-Stokes equations on an half plane. For analytic initial data, existence and uniqueness of the solution are proved using the Abstract Cauchy-Kovalevskaya Theorem in Banach spaces. The time interval of existence is proved to be independent of the viscosity.     

Steady compressible Navier-Stokes flow in a square

We investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain Q∈R². We show existence if a solution that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of the constant flow . In order to show the existence of the solution we adapt the techniques known from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument.     

Three-dimensional blow-up solutions of the Navier-Stokes equations

In this paper we extend the plane blow-up results of Grundy& McLaughlin (1997) to the three-dimensional Navier-Stokes equations.Using a solution structure originally due to Lin we first providenumerical evidence for the existence of blow-up solutions on- < x, z < , 0 y 1 with boundary conditions on y = 0and y = 1 involving derivatives of the velocity components.The formulation enables us to consider plane and radial flowas special cases. Various features of the computations are isolatedand are used to construct a formal asymptotic solution closeto blow-up. We show that the numerical and asymptotic analysesprovide a mutually consistent global picture which supportsthe conclusion that, for the family of problems we considerhere, blow-up in fact can take place in three dimensions butat an inverse linear rate rather than the faster inverse squareof the plane case.     

Asymptotic behaviour of solutions to the Navier-Stokes equations of a two-dimensional compressible flow

In this paper,we are concerned with the asymptotic behaviour of a weak solution to the Navier-Stokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p(e) = a log d(e) for large .Here d > 2,a > 0.We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity.Using properties of this function,we can prove the strong convergence of the density to its limit state.The behaviour of the velocity field and kinetic energy is also briefly discussed.     

Uniqueness of weak solutions of the Navier-Stokes equations

Consider the Navier-Stokes equation with the initial data a ∈ L _σ ²(ℝ ^d ). Let u and v be two weak solutions with the same initial value a. If u satisfies the usual energy inequality and if ∇v ∈ L ²((0, T); (ℝ ^d ) ^d ) where (ℝ ^d ) is the multiplier space, then we have u = v.     

Stationary solutions of the Navier-Stokes equations in periodic tubes

Let be a tubular domain in Rⁿ, n=2,3, with a Lipschitz boundary , invariant with respect to a translation by the vector . It is proven that, for any prescribed real number _o, there exists at least one solution of the nonhomogeneous boundary-value problem for a stationary Navier-Stokes system with a periodic and pressure , having the drop _o over the period. (The exterior forces and the boundary values of the velocity field are assumed to be periodic.) In addition, one proves the existence of a critical nonnegative number ^*, depending only on the geometry of the domain , the viscosity coefficient, the exterior forces and the boundary values of , such that for ¦₀¦>^* the fluid flows along the direction of the decrease of the pressure.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 104–113, 1982.The author is grateful to O. A. Ladyzhenskaya for her interest in the paper.     

Strong solutions of the Navier-Stokes equations in aperture domains

We consider the nonstationary Navier-Stokes equations in an aperture domain Ω⊂R³ consisting of two halfspaces separated by a wall, but connected by a hole in this wall. In this special domain one has to impose an auxiliary condition to single out a unique solution. This can be done by prescribing either the flux through the hole or the pressure drop between the two halfspaces. We construct suitable Stokes operators for both of the auxiliary conditions and show that they generate holomorphic semigroups. Then we prove the existence and uniqueness of solutions as well as a maximal regularity estimate for the Stokes equations subject to one of the auxiliary conditions. For the corresponding Navier-Stokes equations we prove existence and uniqueness of local in time solutions.

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Sunto In questo lavoro consideriamo le equazioni di Navier-Stokes non stazionarie in un dominio con un’apertura, che consiste di due semispazi separati da una parete, ma collegati da un’apertura in quest’ultima. In questo dominio particolare è necessario imporre, per avere un’unica soluzione, una opportuna condizione ausiliaria. Questo può essere fatto sia assegnando il flusso attraverso l’apertura sia prescrivendo il salto di pressione tra i due semispazi. Qui costruiamo degli operatori di Stokes opportuni per ambedue i tipi di condizioni ausiliarie e mostriamo come essi generino semigruppi olomorfi. Dimostriamo, quindi, esistenza e unicità di soluzioni, assieme ad una stima di massima regolarità per le equazioni di Stokes soggette ad una delle condizioni ausiliarie. Per le corrispondenti equazioni di Navier-Stokes, dimostriamo esistenza e unicità di soluzioni locali nel tempo.

     

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.     

Time periodic solutions of compressible Navier-Stokes equations

In this paper, we study the Navier-Stokes equations with a time periodic external force in Rn. We show that a time periodic solution exists when the space dimension n?5 under some smallness assumption. The main idea is to combine the energy method and the spectral analysis for the optimal decay estimates on the linearized solution operator. With the optimal decay estimates, we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem. In addition, we also study the time asymptotic stability of the time periodic solution.     

Self-similar solutions to the isothermal compressible Navier-Stokes equations

^** Email: guo_zhenhua{at}iapcm.ac.cn^*** Email: jiang{at}iapcm.ac.cn We investigate the self-similar solutions to the isothermalcompressible Navier–Stokes equations. The aim of thispaper is to show that there exist neither forward nor backwardself-similar solutions with finite total energy. This generalizesthe results for the incompressible case in Neas, J., Rika, M.& verák, V. (1996, On Leray's self-similar solutionsof the Navier-Stokes equations. Acta. Math., 176, 283–294),and is consistent with the (unproved) existence of regular solutionsglobally in time for the compressible Navier–Stokes equations.     

Special global regular solutions to the Navier-Stokes equations

Ezistence results for global regular solutions to the Navier-Stokes equations, which are close either to two-dimensional or to axially symmetric solutions are presented. Slip boundary conditions are assumed. Moreover, the domains considered are either cylindrical or axially symmetric. Problems with and without inflow-outflow are examined. All proofs can be divided into two steps; (1) long time existence established either by the Leroy Schauder fixed point theorem or by the method of successive approximations; (2) global existence proved by prolongation of a local solution with respcct to time, Bibliography: 32 titles. I dedicate this paper to Vsevolod Alekseevich Solonnikov, the great mathematician and my teacher Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 120–152.     

Global solutions for the Navier-Stokes equations in the rotational framework

The existence of global unique solutions to the Navier-Stokes equations with the Coriolis force is established in the homogeneous Sobolev spaces $\dot{H}^s (\mathbb R ^3)^3$ for $1/2 < s < 3/4$ if the speed of rotation is sufficiently large. This phenomenon is so-called the global regularity. The relationship between the size of initial datum and the speed of rotation is also derived. The proof is based on the space time estimates of the Strichartz type for the semigroup associated with the linearized equations. In the scaling critical space $\dot{H}^{\frac{1}{2}} (\mathbb R ^3)^3$ , the global regularity is also shown.     

Multiple discrete solutions of the incompressible steady-state Navier-Stokes equations

Summary This paper discusses the computation of multiple solutions of various discretizations of the steady state incompressible Navier-Stokes equations. Solution paths (,R) satisfying the discrete system of equationsH(,R)=0, where represents the discrete flow field andR is the Reynolds number, are computed using a pseudo arc-length continuation procedure. For flows over the back end of an axially symmetric body with a cusped tail in a coaxial circular cylinder, the solution paths often exhibit hairpin turning points. Dependence of the paths on the mesh spacing and various selections for the discretizations of the convective and diffusive terms are presented.Dedicated to Professor Ivo Babuka on the occasion of his 60th birthdayThis research was supported in part by the Naval Surface Warfare Center Independent Research Fund, the Naval Sea Systems Command, and the Office for Naval Research under Contracts N001484WR24012 and N0001486R24209, and was performed in part under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract number W-7405-ENG-48. Partial support under contract Number W-7405-ENG-48 was provided by the Applied Mathematical Sciences Program of the Office of Energy Research     

Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders

Summary For the problem of hydrodynamical stability in an infinite cylindrical domain, we investigate all time-periodic solutions, not only spatially periodic ones, when a Hopf bifurcation occurs. When reflection symmetry is present, we show the existence of spatially quasiperiodic flows. We also show the existence of heteroclinic solutions connecting two symmetrically traveling waves that stay at each end of the cylinders (defect solutions). The technique we use rests on (i) a center manifold argument in a space of time-periodic vector fields, (ii) symmetry and normal form arguments for the reduced ordinary differential equation in two dimensions (without reflection symmetry) or in four dimensions (with reflection symmetry), and (iii) the integrability of the associated normal form. It then remains to prove a persistence result when we add the higher-order terms of the vector field.