Stationary Statistical Solutions Research Articles

Limit stationary statistical solutions of stochastic Navier–Stokes–Voigt equation in a 3D thin domain

Limit stationary statistical solutions of stochastic Navier–Stokes–Voigt equation in a 3D thin domain

A principle of maximum entropy for the Navier–Stokes equations

A principle of maximum entropy is proposed in the context of viscous incompressible flow in Eulerian coordinates. The relative entropy functional, defined over the space of L2 divergence-free velocity fields, is maximized relative to alternate measures supported over the energy–enstrophy surface. Since thermodynamic equilibrium distributions are characterized by maximum entropy, connections are drawn with stationary statistical solutions of the incompressible Navier–Stokes equations. Special emphasis is on the correspondence with the final statistics described by Kolmogorov’s theory of fully developed turbulence.

The Rayleigh–Bénard Problem for Compressible Fluid Flows

We consider the physically relevant fully compressible setting of the Rayleigh–Bénard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to gravitational force. Under suitable restrictions imposed on the constitutive relations we show that this open system is dissipative in the sense of Levinson, meaning there exists a bounded absorbing set for any global-in-time weak solution. In addition, global-in-time trajectories are asymptotically compact in suitable topologies and the system possesses a global compact trajectory attractor mathcal {A}. The standard technique of Krylov and Bogolyubov then yields the existence of an invariant measure—a stationary statistical solution sitting on mathcal {A}. In addition, the Birkhoff–Khinchin ergodic theorem provides convergence of ergodic averages of solutions belonging to mathcal {A} a.s. with respect to the invariant measure.

Dynamics for 2D incompressible Navier–Stokes flow coupled with time-dependent Darcy flow

In this paper, we use the method of evolutionary systems introduced by Cheskidov and Foias to describe the existence of global attractor for 2D incompressible Navier–Stokes flow coupled with time-dependent Darcy flow. Furthermore, stationary statistical solutions of this system are constructed from the global attractor.

Ergodic theory for energetically open compressible fluid flows

The ergodic hypothesis is examined for energetically open fluid systems represented by the barotropic Navier–Stokes equations with general inflow/outflow boundary conditions. We show that any globally bounded trajectory generates a stationary statistical solution, which is interpreted as a stochastic process with continuous trajectories supported by the family of weak solutions of the problem. The abstract Birkhoff–Khinchin theorem is applied to obtain convergence (in expectation and a.s.) of ergodic averages for any bounded Borel measurable function of state variables associated to any stationary solution. Finally, we show that validity of the ergodic hypothesis is determined by the behavior of entire solutions (i.e. a solution defined for any t∈R). In particular, the ergodic averages converge for any trajectory provided its ω-limit set in the trajectory space supports a unique (in law) stationary solution.

Globally bounded trajectories for the barotropic Navier–Stokes system with general boundary conditions

We consider the barotropic Navier–Stokes system describing the motion of a viscous compressible fluid interacting with the outer world through general in/out flux boundary conditions. We consider a hard-sphere type pressure EOS and show that all trajectories eventually enter a bounded absorbing set. In particular, the associated limit sets are compact and support a stationary statistical solution.

Properties of Stationary Statistical Solutions of the Three-Dimensional Navier–Stokes Equations

The stationary version of a modified definition of statistical solution for the three-dimensional incompressible Navier–Stokes equations introduced in a previous work is investigated. Particular types of such stationary statistical solutions and their analytical properties are addressed. Results on the support and carriers of these stationary statistical solutions are also given, showing in particular that they are supported on the weak global attractor and are carried by a more regular part of the weak global attractor containing Leray–Hopf weak solutions which are locally strong solutions. Two recurrence-type results related to these measures are also proved.

Convergence of Time Averages of Weak Solutions of the Three-Dimensional Navier–Stokes Equations

Using the concept of stationary statistical solution, which generalizes the notion of invariant measure, it is proved that, in a suitable sense, time averages of almost every Leray–Hopf weak solution of the three-dimensional incompressible Navier–Stokes equations converge as the averaging time goes to infinity. This system of equations is not known to be globally well-posed, and the above result answers a long-standing problem, extending to this system a classical result from ergodic theory. It is also shown that, from a measure-theoretic point of view, the stationary statistical solution obtained from a generalized limit of time averages is independent of the choice of the generalized limit. Finally, any Borel subset of the phase space with positive measure with respect to a stationary statistical solution is such that for almost all initial conditions in that Borel set and for at least one Leray–Hopf weak solution starting with that initial condition, the corresponding orbit is recurrent to that Borel subset and its mean sojourn time within that Borel subset is strictly positive.

A note on statistical solutions of the three-dimensional Navier–Stokes equations: The stationary case

Stationary statistical solutions of the three-dimensional Navier–Stokes equations for incompressible fluids are considered. They are a mathematical formalization of the notion of ensemble average for turbulent flows in statistical equilibrium in time. They are also a generalization of the notion of invariant measure to the case of the three-dimensional Navier–Stokes equations, for which a global uniqueness result is not known to exist and a semigroup may not be well-defined in the classical sense. The two classical definitions of stationary statistical solutions are considered and compared, one of them being a particular case of the other and possessing a number of useful properties. Furthermore, the so-called time-average stationary statistical solutions, obtained as generalized limits of time averages of weak solutions as the averaging time goes to infinity are shown to belong to this more restrictive class. A recurrent type result is also obtained for statistical solutions satisfying an accretion condition. Finally, the weak global attractor of the three-dimensional Navier–Stokes equations is considered, and in particular it is shown that there exists a topologically large subset of the weak global attractor which is of full measure with respect to that particular class of stationary statistical solutions and which has a certain regularity property.

Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit

The Navier-Stokes-Voigt model ofviscoelastic incompressible fluid has been recently proposed as aregularization of the three-dimensional Navier-Stokes equations forthe purpose of direct numerical simulations. Besides the kinematicviscosity parameter, $\nu>0$, this model possesses a regularizingparameter, $\alpha> 0$, a given length scale parameter, so that$\frac{\alpha^2}{\nu}$ is the relaxation time of the viscoelasticfluid. In this work, we derive several statistical properties ofthe invariant measures associated with the solutions of thethree-dimensional Navier-Stokes-Voigt equations. Moreover, we provethat, for fixed viscosity , $\nu>0$, as the regularizing parameter$\alpha$ tends to zero, there exists a subsequence of probabilityinvariant measures converging, in a suitable sense, to a strongstationary statistical solution of the three-dimensionalNavier-Stokes equations, which is a regularized version of thenotion of stationary statistical solutions - a generalization of theconcept of invariant measure introduced and investigated by Foias.This fact supports earlier numerical observations, and provides anadditional evidence that, for small values of the regularizationparameter $\alpha$, the Navier-Stokes-Voigt model can indeed beconsidered as a model to study the statistical properties of thethree-dimensional Navier-Stokes equations and turbulent flows viadirect numerical simulations.

Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations

A new proof of existence of solutions for the three dimensionalsystem of globally modified Navier-Stokes equations introduced in [3] by Caraballo, Kloeden and Real is obtained using a smoother Galerkin scheme. This isthen used to investigate the relationship between invariant measures and statistical solutions of this system in the case of temporally independent forcingterm. Indeed, we are able to prove that a stationary statistical solution is alsoan invariant probability measure under suitable assumptions.

Statistical estimates for channel flows driven by a pressure gradient

We present rigorous estimates for some physical quantities related to turbulent and non-turbulent channel flows driven by a uniform pressure gradient. Such results are based on the concept of stationary statistical solutions, which is related to the notion of ensemble averages for flows in statistical equilibrium. We provide a lower bound estimate for the mean skin friction coefficient and improve on a previous upper bound estimate for the same quantity; both estimates are derived in terms of the Reynolds number. We also present lower and upper bound estimates for the mean rate of energy dissipation, the mean longitudinal bulk velocity (in the direction of the pressure gradient), and the mean kinetic energy in terms of various physical parameters. In particular, we obtain an upper bound related to the energy dissipation law, namely that the mean rate of energy dissipation is essentially bounded by a non-dimensional universal constant times the cube of the mean longitudinal bulk velocity over a characteristic macro-scale length. Finally, we investigate the scale-by-scale energy injection due to the pressure gradient, proving an upper bound estimate for the decrease of this energy injection as the scale length decreases.

Pullback attractors and statistical solutions for 2-D Navier-Stokes equations

In this paper we investigate some relations among the notions of pullback attractor, time-average measure and statistical solution.  &nbsp Using time-averages and Banach generalized limits we construct a family of probability measures $\{\mu_t\}_{t\in \IR}$ on the pullback attractor $\{A(t)\}_{t\in \R}$ of the dynamical system associated with a two-dimensional nonautonomous Navier-Stokes flow in a bounded domain. The measures satisfy supp$\mu_t \subset A(t)$ for all $t\in \R$ and also the corresponding Liouville equation and energy equation. In the autonomous case, they reduce to some time-average measure $\mu$ with support included in the global attractor and being a stationary statistical solution of the Navier-Stokes flow.

Stationary statistical properties of Rayleigh‐Bénard convection at large Prandtl number

AbstractThis is the third in a series of our study of Rayleigh‐Bénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number are related to those of the infinite Prandtl number model for convection that is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for Rayleigh‐Bénard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form which asymptotically agrees with the (optimal) upper bound on Nusselt number for the infinite Prandtl number model for convection. © 2007 Wiley Periodicals, Inc.

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in $$\mathbb R^2$$

We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in ${\mathbb R}^2$. We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enstrophy balance.

Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence

Some rigorous results connected with the conventional statistical theory of turbulence in both the two- and three-dimensional cases are discussed. Such results are based on the concept of stationary statistical solution, related to the notion of ensemble average for turbulence in statistical equilibrium, and concern, in particular, the mean kinetic energy and enstrophy fluxes and their corresponding cascades. Some of the results are developed here in the case of nonsmooth boundaries and a less regular forcing term and for arbitrary stationary statistical solutions.

Navier–Stokes Equations and Turbulence

Preface Acknowledgements 1. Introduction and overview of turbulence 2. Elements of the mathematical theory of the Navier-Stokes equations 3. Finite dimensionality of flows 4. Stationary statistical solutions of the Navier-Stokes equations, time averages and attractors 5. Time-dependent statistical solutions of the Navier-Stokes equations and fully developed turbulence References Index.

Cascade of energy in turbulent flows

A starting point for the conventional theory of turbulence [12–14] is the notion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes [19]. Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so-called cascade of energy, a fundamental mechanism used to explain the Kolmogorov spectrum in three-dimensional turbulent flows. The aim of this Note is to prove this transfer of energy to higher modes in a mathematically rigorous manner, by working directly with the Navier–Stokes equations and stationary statistical solutions obtained through time averages. To the best of our knowledge, this result has not been proved previously; however, some discussions and partly intuitive proofs appear in the literature. See, e.g., [1,2,10,11,16,17,21], and [22]. It is noteworthy that a mathematical framework can be devised where this result can be completely proved, despite the well-known limitations of the mathematical theory of the three-dimensional Navier–Stokes equations. A similar result concerning the transfer of energy is valid in space dimension two. Here, however, due to vorticity constraints not present in the three-dimensional case, such energy transfer is accompanied by a similar transfer of enstrophy to higher modes. Moreover, at low wave numbers, in the spectral region below that of injection of energy, an inverse (from high to low modes) transfer of energy (as well as enstrophy) takes place. These results are directly related to the mechanisms of direct enstrophy cascade and inverse energy cascade which occur, respectively, in a certain spectral range above and below that of injection of energy [1,15]. In a forthcoming article [9] we will discuss conditions for the actual existence of the inertial range in dimension three.

An estimate for the mean square of the difference of velocities for homogeneous statistical solutions of the three-dimensional Navier-Stokes system

Stationary (with respect to t) statistical solutions of the Navier-Stokes system are constructed, in the presence of random external forces which are (statistically) stationary with respect to t. The domains considered are invariant under certain groups of motions (shifts and rotations). The external forces are supposed to be (statistically) invariant with respect to these groups and the constructed statistical solutions are also invariant in the same sense. For the constructed solutions, estimates of the mean square of the difference of velocities at different points are obtained in terms of a certain power of the distance between the points. The exponent depends on the dimension of the symmetry group.