Stochastic Navier Stokes Equations Research Articles

Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations

In this article, we first prove some sufficient conditions guaranteeing the existence of invariant sample measures for random dynamical systems via the approach of global random attractors. Then we consider the two-dimensional incompressible Navier-Stokes equations with additive white noise as an example to show how to check the sufficient conditions for concrete stochastic partial differential equations. Our results generalize the Liouville type theorem to the random case and reveal that the invariance of the sample measures is a particular situation of the random Liouville type theorem.

Remarks on the non-uniqueness in law of the Navier–Stokes equations up to the J.-L. Lions’ exponent

Lions (1959), introduced the Navier–Stokes equations with a viscous diffusion in the form of a fractional Laplacian; subsequently, he (1969, Dunod, Gauthiers-Villars, Paris) claimed the uniqueness of its solution when its exponent is not less than five quarters in case the spatial dimension is three. Following the work of Hofmanová et al. (2019), we prove the non-uniqueness in law for the three-dimensional stochastic Navier–Stokes equations with the viscous diffusion in the form of a fractional Laplacian with its exponent less than five quarters.

Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space dot{mathcal {B}}^{4-2alpha -frac{3}{p}}_{p,r}. Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.

Almost-sure exponential mixing of passive scalars by the stochastic Navier–Stokes equations

We deduce almost-sure exponentially fast mixing of passive scalars advected by solutions of the stochastically-forced 2D Navier–Stokes equations and 3D hyper-viscous Navier–Stokes equations in Td subjected to nondenegenerate Hσ-regular noise for any σ sufficiently large. That is, for all s>0 there is a deterministic exponential decay rate such that all mean-zero Hs passive scalars decay in H−s at this same rate with probability one. This is equivalent to what is known as quenched correlation decay for the Lagrangian flow in the dynamical systems literature. This is a follow-up to our previous work, which establishes a positive Lyapunov exponent for the Lagrangian flow—in general, almost-sure exponential mixing is much stronger than this. Our methods also apply to velocity fields evolving according to finite-dimensional models, for example, Galerkin truncations of Navier–Stokes or the Stokes equations with very degenerate forcing. For all 0≤k<∞, this exhibits many examples of CtkCx∞ random velocity fields that are almost-sure exponentially fast mixers.

Ergodicity for stochastic equations of Navier–Stokes type

In this note we consider a simple example of a finite dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic Navier-Stokes Equations (NSEs), and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant probability measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant probability measure, there exist another one. In particular, the generator of the system is not hypoelliptic.

Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping

Abstract The incompressible 2D stochastic Navier-Stokes equations with linear damping are considered in this paper. Based on some new calculation estimates, we obtain the existence of random attractor and the upper semicontinuity of the random attractors as ε → 0 + \varepsilon \to {0}^{+} on the two-dimensional space.

Large deviation principles of 2D stochastic Navier–Stokes equations with Lévy noises

Taking the consideration of two-dimensional stochastic Navier–Stokes equations with multiplicative Lévy noises, where the noises intensities are related to the viscosity, a large deviation principle is established by using the weak convergence method skillfully, when the viscosity converges to 0. Due to the appearance of the jumps, it is difficult to close the energy estimates and obtain the desired convergence. Hence, one cannot simply use the weak convergence approach. To overcome the difficulty, one introduces special norms for new arguments and more careful analysis.

Recurrent motion of stochastic Navier–Stokes equations with Coriolis force

AbstractIn this paper, we consider recurrent motion for stochastic Navier–Stokes equations in rotational framework. We prove the unique existence of time periodic, almost periodic and almost automorphic solutions for the stochastic partial differential equations.

Large deviations principle for the invariant measures of the 2D stochastic Navier–Stokes equations with vanishing noise correlation

We study the two-dimensional incompressible Navier–Stokes equation on the torus, driven by Gaussian noise that is white in time and colored in space. We consider the case where the magnitude of the random forcing \(\sqrt{{\epsilon }}\) and its correlation scale \(\delta ({\epsilon })\) are both small. We prove a large deviations principle for the solutions, as well as for the family of invariant measures, as \({\epsilon }\) and \(\delta ({\epsilon })\) are simultaneously sent to 0, under a suitable scaling.

Space-time Euler discretization schemes for the stochastic 2D Navier–Stokes equations

We prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier–Stokes equations on the torus subject to a random perturbation converges in \(L^2(\varOmega )\), and describe the rate of convergence for an \(H^1\)-valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier–Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The speed of the \(L^2(\varOmega )\)-convergence depends on the diffusion coefficient and on the viscosity parameter. In case of Scott–Vogelius mixed elements and for an additive noise, the convergence is polynomial.

Statistics of SPDEs: From Linear to Nonlinear

We study statistical inference for stochastic partial differential equations (SPDEs). Though inference linear SPDEs have been studied well (with lot of problems still remain to be investigated) in the last two decades, inference for nonlinear SPDEs is in its infancy. The inference methods use both inference for finite-dimensional diffusions and inference for classical i.i.d. sequences. Solving 2D Navier-Stokes equation is one of the challenging problem of the last century. However, with additive white noise, the equation has a strong solution. We estimate the viscosity coefficient of the 2D stochastic Navier-Stokes (SNS) equation by minimum contrast method. We show $n$ consistency in contrast to $\sqrt n$ consistency in the classical i.i.d. case where $n$ is the number of observations. We consider both continuous and discrete observations in time. We also obtain the Berry-Esseen bounds. Then we estimate and control the Type I and Type II error of a simple hypothesis testing problem of the viscosity coefficient of the SNS equation. We study a class of rejection regions and provide thresholds that guarantee that the statistical errors are smaller than the given upper bound. The tests are of likelihood ratio type. The proofs are based on the large deviation bounds. Finally we give Monte Carlo test procedure for simulated data.

A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations

In this article, we review our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations. This hallmark of chaos has long been observed in these models, however, no mathematical proof had been made for either deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an $L^1$-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies H\"ormander's condition. We review the recent contributions of the first and third author on the verification of this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio. Finally, we briefly contrast this work with our earlier work on Lagrangian chaos in the stochastic Navier-Stokes equations. We end the review with a discussion of some open problems.

Numerical analysis of fully discrete finite element methods for the stochastic Navier-Stokes equations with multiplicative noise

Previous work on the stability and convergence analysis of the finite element methods for the deterministic Navier-Stokes equations was carried out under the uniqueness condition. In this paper, the corresponding results of fully discrete finite element method are developed for the stochastic Navier-Stokes equations with multiplicative noise. The stability and error estimates for velocity in expectation are rigorously proved, and the L2-norm in a time-averaged version for pressure is provided with convergence rates in spatial and temporal form O(Δt+hΔt). Finally, numerical experiments are given to illustrate the features of the proposed numerical method and verify the theoretical claims.

A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier\u2013Stokes Equations

We analyze the control problem of the stochastic Navier–Stokes equations in multi-dimensional domains considered in Benner and Trautwein (Math Nachr 292(7):1444–1461, 2019) restricted to noise terms defined by a Q-Wiener process. The cost functional related to this control problem is nonconvex. Using a stochastic maximum principle, we derive a necessary optimality condition to obtain explicit formulas the optimal controls have to satisfy. Moreover, we show that the optimal controls satisfy a sufficient optimality condition. As a consequence, we are able to solve uniquely control problems constrained by the stochastic Navier–Stokes equations especially for two-dimensional as well as for three-dimensional domains.

Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in linear parabolic equations for vorticity. Probabilistic representations for solutions of these linear equations are given. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot--Savart-type law. We show that the approximation is divergent free and of first order. The results are extended to two-dimensional stochastic Navier-Stokes equations with additive noise, where, in particular, we prove the first mean-square convergence order of the vorticity approximation.

Global existence for the stochastic Navier–Stokes equations with small $$L^{p}$$ data

Global existence for the stochastic Navier–Stokes equations with small $$L^{p}$$ data

Maximum principle for optimal control of SPDEs with locally monotone coefficients

ABSTRACT The aim of this paper is to derive a maximum principle for a control problem governed by a stochastic partial differential equation (SPDE) with locally monotone coefficients. To reach our goal we adapt the method which uses the relation between backward stochastic partial differential equation (BSPDE) and the maximum principle. In particular, necessary conditions for optimality for this stochastic optimal control problem are obtained. In spite of the fact that the method used here was used by several authors before, our adaptation is not immediate. It applies a trick which is used to get estimates for solutions of SPDE with Locally Monotone Coefficients as in the proof of the Lemmas 5.1 and 5.3. This adaptation permits us to apply our results to get a maximum principle for the optimal control to the cases when the system is governed by the 2D stochastic Navier-Stokes equation and by a stochastic Burgers' equation.

Convergence rates for the numerical approximation of the 2D stochastic Navier\u2013Stokes equations

We study stochastic Navier–Stokes equations in two dimensions with respect to periodic boundary conditions. The equations are perturbed by a nonlinear multiplicative stochastic forcing with linear growth (in the velocity) driven by a cylindrical Wiener process. We establish convergence rates for a finite-element based space-time approximation with respect to convergence in probability (where the error is measured in the L^infty _tL^2_xcap L^2_tW^{1,2}_x-norm). Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from Carelli and Prohl (SIAM J Numer Anal 50(5):2467–2496, 2012) where the convergence rate in time is only (almost) 1/4. Our approach is based on a careful analysis of the pressure function using a stochastic pressure decomposition.

Uniform random attractors for 2D non-autonomous stochastic Navier-Stokes equations

In this paper, we first establish the existence of uniform random attractor for 2D stochastic Navier-Stokes equation in H with deterministic non-autonomous external force being normal in Lloc2(R;V′), which is the measurable minimal compact set and uniformly attracts bounded random set in H in the sense of pullback. We also show that uniform random attractor with respect to the deterministic non-autonomous functions belonging to some symbol space coincides with uniform random attractor with respect to the initial time. Then we show that the uniform random attractor for the equation under consideration has regularity property in V when deterministic non-autonomous external force being normal in Lloc2(R;H).

Stochastic fluid dynamics simulations of the velocity distribution in protoplasmic streaming

Protoplasmic streaming in plant cells is directly visible in the cases of Chara corallina and Nitella flexilis, and this streaming is understood to play a role in the transport of biological materials. For this reason, related studies have focused on molecular transportation from a fluid mechanics viewpoint. However, the experimentally observed distribution of the velocity along the flow direction x, which exhibits two peaks at Vx = 0 and at a finite Vx(≠0), remains to be studied. In this paper, we numerically study whether this behavior of the flow field can be simulated by a 2D stochastic Navier–Stokes (NS) equation for Couette flow in which a random Brownian force is assumed. We present the first numerical evidence that these peaks are reproduced by the stochastic NS equation, which implies that the Brownian motion of the fluid particles plays an essential role in the emergence of these peaks in the velocity distribution. We also find that the position of the peak at Vx(≠0) moves with the variation in the strength D of the random Brownian force, which also changes depending on physical parameters such as the kinematic viscosity, boundary velocity, and diameter of the plant cells.