Stochastic Navier Stokes Equations Research Articles

Numerical approximation of the stochastic Navier–Stokes equations through artificial compressibility

Numerical approximation of the stochastic Navier–Stokes equations through artificial compressibility

Chaos in Stochastic 2d Galerkin-Navier–Stokes

We prove that all Galerkin truncations of the 2d stochastic Navier–Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies N≥392\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 392$$\\end{document}. By “chaotic” we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander’s condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using (a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and (b) computational algebraic geometry executed by Maple in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit.

Sharp Nonuniqueness of Solutions to Stochastic Navier–Stokes Equations

Sharp Nonuniqueness of Solutions to Stochastic Navier–Stokes Equations

On initial-boundary value problem of the stochastic Navier–Stokes equations in the half space

We study the initial-boundary value problem of the stochastic Navier–Stokes equations in half-space. We prove the existence of weak solutions in standard Besov space-valued random processes when the initial data belong to the critical Besov space.

Stochastic Navier–Stokes Equations for Turbulent Flows in Critical Spaces

In this paper we study the stochastic Navier–Stokes equations on the d-dimensional torus with transport noise, which arise in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local well-posedness in the critical case Bq,pd/q-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^{d/q-1}_{q,p}$$\\end{document} for q∈[2,2d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\in [2,2d)$$\\end{document} and p large enough. Moreover, we obtain new regularization results for solutions, and new blow-up criteria which can be seen as a stochastic version of the Serrin blow-up criteria. The latter is used to prove global well-posedness with high probability for small initial data in critical spaces in any dimensions d⩾2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\geqslant 2$$\\end{document}. Moreover, for d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}, we obtain new global well-posedness results and regularization phenomena which unify and extend several earlier results.

Inviscid limit for stochastic Navier-Stokes equations under general initial conditions

We consider in a smooth and bounded two dimensional domain the convergence in the L2 norm, uniformly in time, of the solution of the stochastic Navier-Stokes equations with additive noise and no-slip boundary conditions to the solution of the corresponding Euler equations. We prove, under general regularity on the initial conditions of the Euler equations, that assuming the dissipation of the energy of the solution of the Navier-Stokes equations in a Kato type boundary layer, then the inviscid limit holds.

Well-posedness and invariant measures for 2D stochastic Oldroyd model of order one with pure jumps

In this paper, we consider the stochastic two-dimensional viscoelastic fluid flow equations driven by a multiplicative Lévy noise, arising from the Oldroyd model for the non-Newtonian fluid flows. The system carries the natural Lipschitz condition on balls and linear growth assumptions on the jump coefficient. We first prove the existence and uniqueness of a global strong solution by using stopping time technique and Banach fixed point theorem. Meanwhile, we overcome the difficulties posed by the jump and memory term, the latter makes some convergence findings more natural. By demonstrating the exponential stability of strong solutions, we establish the existence of a unique invariant measure for the stochastic system. Numerical results provide further support for these theoretical findings, the dissipative velocity of solution is more rapid and rigorous when compared with the stochastic Navier-Stokes equation.

Asymptotic autonomy of random attractors for non-autonomous stochastic Navier-Stokes equations on bounded domains

Asymptotic autonomy of random attractors for non-autonomous stochastic Navier-Stokes equations on bounded domains

Global existence for perturbations of the 2D stochastic Navier–Stokes equations with space-time white noise

We prove global in time well-posedness for perturbations of the 2D stochastic Navier–Stokes equations ∂tu+u·∇u=Δu-∇p+ζ+ξ,u(0,·)=u0,div(u)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u + u \\cdot \ abla u= & {} \\Delta u - \ abla p + \\zeta + \\xi \\;, \\qquad u (0, \\cdot ) = u_{0} \\;,\\\\ {\ ext {div}}(u)= & {} 0 \\;, \\end{aligned}$$\\end{document}driven by additive space-time white noise ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\xi $$\\end{document}, with perturbation ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\zeta $$\\end{document} in the Hölder–Besov space C-2+3κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}^{-2 + 3\\kappa } $$\\end{document}, periodic boundary conditions and initial condition u0∈C-1+κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ u_{0} \\in \\mathcal {C}^{-1 + \\kappa } $$\\end{document} for any κ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\kappa >0 $$\\end{document}. The proof relies on an energy estimate which in turn builds on a dynamic high-low frequency decomposition and tools from paracontrolled calculus. Our argument uses that the solution to the linear equation is a log\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\log $$\\end{document}–correlated field, yielding a double exponential growth bound on the solution. Notably, our method does not rely on any explicit knowledge of the invariant measure to the SPDE, hence the perturbation ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\zeta $$\\end{document} is not restricted to the Cameron–Martin space of the noise, and the initial condition may be anticipative. Finally, we introduce a notion of weak solution that leads to well-posedness for all initial data u0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ u_{0}$$\\end{document} in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ L^{2} $$\\end{document}, the critical space of initial conditions.

Three-dimensional stochastic Navier–Stokes equations with Markov switching

A finite-state Markov chain is introduced in the noise terms of the three-dimensional stochastic Navier–Stokes equations in order to allow for transitions between two types of multiplicative noises. We call such systems as stochastic Navier–Stokes equations with Markov switching. To solve such a system, a family of regularized stochastic systems is introduced. For each such regularized system, the existence of a unique strong solution (in the sense of stochastic analysis) is established by the method of martingale problems and pathwise uniqueness. The regularization is removed in the limit by obtaining a weakly convergent sequence from the family of regularized solutions, and identifying the limit as a solution of the three-dimensional stochastic Navier–Stokes equation with Markov switching.

Error Analysis for 2D Stochastic Navier–Stokes Equations in Bounded Domains with Dirichlet Data

AbstractWe study a finite-element based space-time discretisation for the 2D stochastic Navier–Stokes equations in a bounded domain supplemented with no-slip boundary conditions. We prove optimal convergence rates in the energy norm with respect to convergence in probability, that is convergence of order (almost) 1/2 in time and 1 in space. This was previously only known in the space-periodic case, where higher order energy estimates for any given (deterministic) time are available. In contrast to this, estimates in the Dirichlet-case are only known for a (possibly large) stopping time. We overcome this problem by introducing an approach based on discrete stopping times. This replaces the localised estimates (with respect to the sample space) from earlier contributions.

A short remark on inviscid limit of the stochastic Navier–Stokes equations

In this article, we study the inviscid limit of the stochastic incompressible Navier–Stokes equations in three-dimensional space. We prove that a subsequence of weak martingale solutions of the stochastic incompressible Navier–Stokes equations converges strongly to a weak martingale solution of the stochastic incompressible Euler equations in the periodic domain under the well-accepted hypothesis, namely Kolmogorov hypothesis (K41).

Large deviations for the two-time-scale stochastic convective Brinkman-Forchheimer equations

The convective Brinkman-Forchheimer (CBF) equations are employed to characterize the motion of incompressible fluid in a saturated porous medium. This work investigates the small noise asymptotic of two-time-scale stochastic CBF equations in two and three dimensional bounded domains. More precisely, we establish a Wentzell-Freidlin type large deviation principle for stochastic partial differential equations that have slow and fast time-scales. The slow component is the stochastic CBF equations in two or three dimensions perturbed by a small multiplicative Gaussian noise, while the fast component is a stochastic reaction-diffusion equation with damping. The results are obtained by using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis, Khasminkii's time discretization approach and stopping time arguments. In particular, the findings from this study are also applicable to two-dimensional stochastic Navier-Stokes equations.

Enhanced Dissipation for Stochastic Navier–Stokes Equations with Transport Noise

Enhanced Dissipation for Stochastic Navier–Stokes Equations with Transport Noise

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.

Asymptotically Autonomous Robustness in Probability of Random Attractors for Stochastic Navier-Stokes Equations on Unbounded Poincaré Domains

Asymptotically Autonomous Robustness in Probability of Random Attractors for Stochastic Navier-Stokes Equations on Unbounded Poincaré Domains

Numerical ergodicity of two dimensional stochastic Navier-Stokes equations with Gaussian noise

Numerical ergodicity of two dimensional stochastic Navier-Stokes equations with Gaussian noise

Large Deviation Principle for the Two-dimensional Stochastic Navier-Stokes Equations with Anisotropic Viscosity

Large Deviation Principle for the Two-dimensional Stochastic Navier-Stokes Equations with Anisotropic Viscosity

Ergodicity for three-dimensional stochastic Navier–Stokes equations with Markovian switching

Asymptotic behavior of the three-dimensional stochastic Navier-Stokes equations with Markovian switching for additive noises is studied for an incompressible fluid flow in a bounded domain. The objective of this article is to prove the existence and identification of a stationary measure for such stochastic systems. To achieve this goal, a family of regularized equations is introduced in order to investigate the asymptotic behavior of their solutions. The existence of an ergodic measure for the regularized system is established by the Krylov-Bogolyubov method. The existence and identification of a stationary measure corresponding to the original system are obtained as a sequential limit point of the family of the ergodic measures that correspond to the regularized systems. The main result of this work is that under mild integrability conditions on the initial data and external force, there exists a stationary measure for the solution of the three-dimensional stochastic Navier-Stokes equations with Markovian switching.

Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability

UDC 519.21 We initiate the investigation of stochastic Navier–Stokes variational inequalities involving unilateral boundary conditions and nonlinear forcings driven by Wiener processes for which we establish the existence of a probabilistic weak (or martingale) solution. Our approach involves an intermediate penalized problem whose weak solution is obtained by means of Galerkin's method in combination with some analytic and probabilistic compactness results. The required probabilistic weak solution of the stochastic Navier–Stokes variational inequality is consecutively obtained through the limit transition in the penalized problem. The main result is new for stochastic Navier–Stokes variational inequalities. It is a stochastic counterpart of the work of Brezis on deterministic Navier–Stokes variational inequalities and generalizes several previous results on stochastic Navier-Stokes equations to stochastic Navier–Stokes variational inequalities with unilateral boundary conditions.