3D Stochastic Navier Stokes Equations Research Articles

Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm

Abstract This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on ${\mathbb{R}}^{+}$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain ‘weak Harris theorems’. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting stochastic partial differential equation examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier–Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on ${\mathbb{R}}^{+}$. To develop these numerical analysis results, we provide a refinement of $L^{2}_{x}$ accuracy bounds in comparison to the existing literature, which are results of independent interest.

A lower bound estimate of life span of solutions to stochastic 3D Navier–Stokes equations with convolution-type noise

In this paper, we investigate the stochastic 3D Navier–Stokes equations perturbed by linear multiplicative Gaussian noise of convolution type by transformation to random PDEs. We focus on obtaining bounds from below for the life span associated with regular initial data. The key point of the proof is the fixed point argument.

Accessibility of SPDEs driven by pure jump noise and its applications

In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump (possibly degenerate) noise, including stochastic 2D Navier-Stokes equations, stochastic Burgers equations, stochastic singular p p -Laplace equations, and stochastic fast diffusion equations. As a further application, we establish the ergodicity of stochastic singular p p -Laplace equations and stochastic fast diffusion equations driven by additive pure jump noise, and we remark that the driving noises could be Compound Poisson processes or Lévy processes with heavy tails.

Dynamics of locally monotone stochastic evolution equations

This paper is concerned with the dynamics of the locally monotone stochastic evolution equations in Bochner spaces. We first prove the well-posedness of the abstract stochastic evolution equations with locally Lipschitz nonlinear diffusion terms. Then we establish a mean random dynamical system and prove the existence and uniqueness of weak pullback mean random attractors generated by the mean random dynamical system. Furthermore, we prove the existence of invariant measures for the locally monotone model. As applications, the related dynamics for several stochastic models such as stochastic semilinear equations, stochastic Burgers equation and stochastic 2D Navier-Stokes equations are established.

Space-Time Approximation of Local Strong Solutions to the 3D Stochastic Navier–Stokes Equations

Abstract We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution.

Deep Latent Regularity Network for Modeling Stochastic Partial Differential Equations

Stochastic partial differential equations (SPDEs) are crucial for modelling dynamics with randomness in many areas including economics, physics, and atmospheric sciences. Recently, using deep learning approaches to learn the PDE solution for accelerating PDE simulation becomes increasingly popular. However, SPDEs have two unique properties that require new design on the models. First, the model to approximate the solution of SPDE should be generalizable over both initial conditions and the random sampled forcing term. Second, the random forcing terms usually have poor regularity whose statistics may diverge (e.g., the space-time white noise). To deal with the problems, in this work, we design a deep neural network called \emph{Deep Latent Regularity Net} (DLR-Net). DLR-Net includes a regularity feature block as the main component, which maps the initial condition and the random forcing term to a set of regularity features. The processing of regularity features is inspired by regularity structure theory and the features provably compose a set of basis to represent the SPDE solution. The regularity features are then fed into a small backbone neural operator to get the output. We conduct experiments on various SPDEs including the dynamic $\Phi^4_1$ model and the stochastic 2D Navier-Stokes equation to predict their solutions, and the results demonstrate that the proposed DLR-Net can achieve SOTA accuracy compared with the baselines. Moreover, the inference time is over 20 times faster than the traditional numerical solver and is comparable with the baseline deep learning models.

Nonuniqueness in law of stochastic 3D Navier–Stokes equations

We consider the stochastic Navier–Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits us to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail to satisfy the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval [0,\infty) . Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, nonuniqueness in law holds on an arbitrary time interval [0,T] , T>0 .

Stochastic Integral Evolution Equations with Locally Monotone and Non-Lipschitz Coefficients

In this work the existence and uniqueness of strong solutions are established for a class of stochastic integral evolution equations with locally monotone and non-Lipschitz coefficients. In particular, our results can be applied to a large class of stochastic partial differential equation models with hereditary or memory effects such as quasilinear SPDEs like stochastic porous medium equations and semilinear SPDEs like stochastic 2D Navier—Stokes equations with hereditary or memory terms.

Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier–Stokes equations: Existence and nonuniqueness

We are concerned with the three-dimensional incompressible Navier–Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in L2 we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in particular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows.

The continuity, regularity and polynomial stability of mild solutions for stochastic 2D-Stokes equations with unbounded delay driven by tempered fractional Gaussian noise

We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order [Formula: see text] driven by a tempered fractional Brownian motion (TFBM) [Formula: see text] with [Formula: see text] and [Formula: see text]. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of [Formula: see text]th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of [Formula: see text], [Formula: see text] or [Formula: see text], [Formula: see text]. In particular, we obtain [Formula: see text]th moment Hölder regularity in time and [Formula: see text]th polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise.

Averaging Principles for Stochastic 2D Navier–Stokes Equations

Averaging Principles for Stochastic 2D Navier–Stokes Equations

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.

Approximations of the stochastic 3D Navier–Stokes equations with damping

Approximations of the stochastic 3D Navier–Stokes equations with damping

Averaging principle for slow–fast stochastic 2D Navier–Stokes equation driven by Lévy noise

This paper investigates the multiscale stochastic 2D Navier–Stokes equation driven by multiplicative Lévy noise. To be more precise, we establish the strong averaging principle for the stochastic 2D Navier–Stokes equation driven by Lévy noise, which involves a fast time scale component governed by a stochastic reaction‐diffusion equation driven by Lévy noise. The Khasminkii's time discretization approach and the technique of stopping time play the important roles in our proof.

Stability of stochastic 2D Navier-Stokes equations with memory and Poisson jumps

The objective of this paper is to study the stability of the weak solutions of stochastic 2D Navier-Stokes equations with memory and Poisson jumps. The asymptotic stability of the stochastic Navier-Stoke equation as a semilinear stochastic evolution equation in Hilbert spaces is obtained in both mean square and almost sure senses. Our results can extend and improve some existing ones.

Deterministic and stochastic 2D Navier-Stokes equations with anisotropic viscosity

In this paper, we investigate both deterministic and stochastic 2D Navier-Stokes equations with anisotropic viscosity. For the deterministic case, we prove the global well-posedness of the system with initial data in the anisotropic Sobolev space H˜0,1. For the stochastic case, we obtain existence of the martingale solutions and pathwise uniqueness of the solutions, which imply existence of the probabilistically strong solution to this system by the Yamada-Watanabe Theorem.

Asymptotic log-Harnack inequality and applications for stochastic 2D hydrodynamical-type systems with degenerate noise

In this paper, an asymptotic log-Harnack inequality and some consequent properties are established via the asymptotic coupling method for a class of stochastic 2D hydrodynamical-type systems driven by degenerate noise. The main results are applicable to the stochastic 2D Navier–Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 2D Boussinesq equations, stochastic 2D magnetic Benard problem, stochastic 3D Leray- $$\alpha $$ model and also stochastic shell models of turbulence in the degenerate noise case.

Stochastic Navier-Stokes Equations and Related Models

Regularization by noise for certain classes of fluid dynamic equations, a theme dear to Giuseppe Da Prato (see G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl., 2003), is reviewed focusing on 3D Navier-Stokes equations and dyadic models of turbulence.

Sufficient Conditions for Dual Cascade Flux Laws in the Stochastic 2d Navier–Stokes Equations

We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws capturing the energy flux to large scales and enstrophy flux to small scales for statistically stationary, forced-dissipated 2d Navier-Stokes equations in the large-box limit. These laws should be regarded as 2d turbulence analogues of the $4/5$ law in 3d turbulence, predicting a constant flux of energy and enstrophy (respectively) through the two inertial ranges in the dual cascade of 2d turbulence. Conditions implying only one of the two cascades are also obtained, as well as compactness criteria which show that the provided sufficient conditions are not far from being necessary. The specific goal of the work is to provide the weakest characterizations of the '0-th laws' of 2d turbulence in order to make mathematically rigorous predictions consistent with experimental evidence.

Random attractors for locally monotone stochastic partial differential equations

We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-α model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard type equations, stochastic Kuramoto-Sivashinsky type equations, stochastic generalized porous media equations and stochastic generalized p-Laplace equations.