Stochastic Navier Research Articles

The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier–Stokes equations driven by Levy processes

Let D be a bounded or unbounded open domain of 2-dimensional Euclidean space R 2 . If the boundary ∂ D = Γ exists, then we assume that the boundary is smooth. In this paper assuming that the kinematic viscosity ν > 0 is large enough, we discuss the existence and exponential stability of energy solutions to the following 2-dimensional stochastic functional Navier–Stokes equation perturbed by the Levy process: { d X ( t ) = [ ν Δ X ( t ) + 〈 X ( t ) , ∇ 〉 X ( t ) + f ( t , X ( t ) ) + F ( t , X t ) − ∇ p ] d t d X ( t ) = + g ( t , X ( t ) ) d W ( t ) + ∫ U k ( t , X ( t ) , y ) q ( d t d y ) , d i v X = 0 in [ 0 , ∞ ) × D , where X ( t , x ) = φ ( t , x ) is the initial function for x ∈ D and t ∈ [ − r , 0 ] with r > 0 . It is assumed that f , g , F and k satisfy the Lipschitz condition and the linear growth condition. If there exists the boundary ∂ D, then X ( t , x ) = 0 on [ 0 , ∞ ) × ∂ D .

Capillary waves of compressible fluids

The interplay of thermal noise and molecular forces is responsible for surprisingfeatures of liquids on sub-micrometer lengths—in particular at interfaces. Notonly does the surface tension depend on the size of an applied distortion andnanoscopic thin liquid films dewet faster than would be expected from hydrodynamics,but also the dispersion relation of capillary waves differ at the nanoscale fromthe familiar macroscopic behavior. Starting with the stochastic Navier–Stokesequation we study the coupling of capillary waves to acoustic surface waves which ispossible in compressible fluids. We find propagating ‘acoustic–capillary waves’ atnanometer wavelengths where in incompressible fluids capillary waves are overdamped.

Ergodicity of stochastic 2D Navier–Stokes equation with Lévy noise

In this paper we deal with the 2D Navier–Stokes equation perturbed by a Lévy noise force whose white noise part is non-degenerate and that the intensity measure of Poisson measure is σ-finite. Existence and uniqueness of invariant measure for this equation is obtained, two main properties of the Markov semigroup associated with this equation are proved. In other words, strong Feller property and irreducibility hold in the same space.

Probabilistic evaluation of the aerodynamic properties of a bridge deck

The study of the probabilistic response of wind-excited structures and their reliability analysis need to take into account for the influence of a large number of uncertain parameters ascribed to meteorological data, structural characteristics and aerodynamic/aeroelastic behaviour. Although the latter one plays a key role in the reliability analysis and it shows high sensitivity to a number of parameters, the probabilistic description of the aerodynamic and aeroelastic forces remains scarce in literature. The present work aims at giving a contribution to the probabilistic evaluation of the aerodynamic properties of a bluff body, namely a bare bridge deck with trapezoidal cross section. The section lower edge degree of sharpness and the oncoming turbulence integral length scale are retained as random variables. The computational solution of the resulting stochastic Navier–Stokes equations is performed by means of the adaptive form of the Multi-Element generalized Polynomial Chaos. As a result, the stochastic aerodynamic coefficients are described by their approximate probability density functions, which are related in turn to the aerodynamic regimes recognised for the case study.

Asymptotic behaviors of stochastic two-dimensional Navier–Stokes equations with finite memory

The stochastic 2D Navier–Stokes equations with finite memory are studied. For the differentiable memory function, the almost sure exponential stability of the weak solution is shown by employing a non-negative semimartingale convergence argument. For the nondifferentiable memory function, the exponential stability in mean square for the weak solution is proved by using the differentiability property of the moment function.

Martingale solutions and Markov selection of stochastic 3D Navier–Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier–Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.

Conservation of Total Vorticity for a 2D Stochastic Navier Stokes Equation

We consider N point vortices whose positions satisfy a stochastic ordinary differential equation on ℝ2N perturbed by spatially correlated Brownian noise. The associated signed point measure‐valued empirical process turns out to be a weak solution to a stochastic Navier‐Stokes equation (SNSE) with a state‐dependent stochastic term. As the number of vortices tends to infinity, we obtain a smooth solution to the SNSE, and we prove the conservation of total vorticity in this continuum limit.

Ergodicity of the 3D stochastic Navier–Stokes equations driven by mildly degenerate noise

We prove that any Markov solution to the 3D stochastic Navier–Stokes equations driven by a mildly degenerate noise (i.e. all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.

Weak solutions to stochastic 3D Navier–Stokes- α model of turbulence: α-Asymptotic behavior

We study the asymptotic behavior of weak solutions to the stochastic 3D Navier–Stokes- α model as α approaches zero. The main result provides a new construction of the weak solutions of stochastic 3D Navier–Stokes equations as approximations by sequences of solutions of the stochastic 3D Navier–Stokes- α model.

DENSITY DEPENDENT STOCHASTIC NAVIER–STOKES EQUATIONS WITH NON-LIPSCHITZ RANDOM FORCING

In this work, we investigate the question of existence of weak solutions to the density dependent stochastic Navier–Stokes equations. The noise considered contains functions which depend nonlinearly on the velocity and which do not satisfy the Lipschitz condition. Furthermore, the initial density is allowed to vanish. We introduce a suitable notion of probabilistic weak solution for the problem and prove its existence.

Asymptotic behavior of solutions of stochastic evolution equations for second grade fluids

In this Note we show that under suitable conditions on the data we can construct a sequence of solutions of the stochastic second grade fluid that converges to the probabilistic strong solution of the stochastic Navier–Stokes equations when the stress modulus α tends to zero.

Existence of martingale and stationary suitable weak solutions for a stochastic Navier–Stokes system

The existence of suitable weak solutions of 3D Navier–Stokes equations, driven by a random body force, is proved. These solutions satisfy a local balance of energy. Existence of statistically stationary solutions is also proved.

Existence, uniqueness and statistical theory of turbulent solutions of the stochastic Navier--Stokes equation, in three dimensions, an overview

We discuss the proofs of the existence and uniqueness of solutions of the Navier--Stokes equation driven with additive noise in three dimensions, in the presence of a strong uni-directional mean flow with some rotation. We also discuss how the existence of a unique invariant measure is established and the properties of this measure are described. The invariant measure is used to prove Kolmogorov's scaling in 3-dimensional turbulence including the celebrated $-5/3$ power law for the decay of the power spectrum of a turbulent 3-dimensional flow. Then we briefly describe the mathematical proof of Kolmogorov's statistical theory of turbulence.

Stochastic Volterra equations in Banach spaces and stochastic partial differential equation

In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier–Stokes equations are also investigated.

Dynamics of stochastic 2 D Navier–Stokes equations

In this paper, we study the dynamics of a two-dimensional stochastic Navier–Stokes equation on a smooth domain, driven by linear multiplicative white noise. We show that solutions of the 2 D Navier–Stokes equation generate a perfect and locally compacting C 1 , 1 cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium/stationary solution. We give sufficient conditions on the parameters of the Navier–Stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space.

On the small time asymptotics of the two-dimensional stochastic Navier–Stokes equations

Dans cet article, nous etablissons un principe de grandes deviations en temps petit pour l’equation de Navier–Stokes bi-dimensionnelle stochastique conduite par un bruit multiplicatif. Celui-ci necessite non seulement l’etude d’un bruit faible, mais aussi la comprehension des effets de derives petites mais non bornees et non lineaires.

Anomalous scaling of a passive vector advected by the Navier–Stokes velocity field

Using the field theoretic renormalization group and the operator-product expansion, the model of a passive vector field (a weak magnetic field in the framework of the kinematic MHD) advected by the velocity field which is governed by the stochastic Navier–Stokes equation with the Gaussian random stirring force δ-correlated in time and with the correlator proportional to k4−d−2ε is investigated to the first order in ε (one-loop approximation). It is shown that the single-time correlation functions of the advected vector field have anomalous scaling behavior and the corresponding exponents are calculated in the isotropic case, as well as in the case with the presence of large-scale anisotropy. The hierarchy of the anisotropic critical dimensions is briefly discussed and the persistence of the anisotropy inside the inertial range is demonstrated on the behavior of the skewness and hyperskewness (dimensionless ratios of correlation functions) as functions of the Reynolds number Re. It is shown that even though the present model of a passive vector field advected by the realistic velocity field is mathematically more complicated than, on one hand, the corresponding models of a passive vector field advected by ‘synthetic’ Gaussian velocity fields and, on the other hand, than the corresponding model of a passive scalar quantity advected by the velocity field driven by the stochastic Navier–Stokes equation, the final one-loop approximate asymptotic scaling behavior of the single-time correlation or structure functions of the advected fields of all models are defined by the same anomalous dimensions (up to normalization).

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.

A Newton method for the resolution of steady stochastic Navier–Stokes equations

We present a Newton method to compute the stochastic solution of the steady incompressible Navier–Stokes equations with random data (boundary conditions, forcing term, fluid properties). The method assumes a spectral discretization at the stochastic level involving a orthogonal basis of random functionals (such as Polynomial Chaos or stochastic multi-wavelets bases). The Newton method uses the unsteady equations to derive a linear equation for the stochastic Newton increments. This linear equation is subsequently solved following a matrix-free strategy, where the iterations consist in performing integrations of the linearized unsteady Navier–Stokes equations, with an appropriate time scheme to allow for a decoupled integration of the stochastic modes. Various examples are provided to demonstrate the efficiency of the method in determining stochastic steady solution, even for regimes where it is likely unstable.

On Strong $H_{2}^{1}$-Solutions of Stochastic Navier–Stokes Equation in a Bounded Domain

We prove the existence and uniqueness of strong local $H_{2}^{1}$-solutions to stochastic Navier–Stokes equation in a bounded domain. The solution is global in two dimensions (2D).MSC codes60H1535Q3076D03MSC codesstochastic Navier–Stokes equationstrong solutions