Stochastic tamed Navier-Stokes equations on ${\mathbb{R}}^3$:existence, uniqueness of solution and existence of an invariant measure

Abstract

Röckner and Zhang in [27] proved the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space and for the periodic boundary case using a result from [31]. In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the $L^4-$norm of the solution from the torus to ${\mathbb{R}}^3$, see Lemma 5.1 and thus establish the existence of an invariant measure on ${\mathbb{R}}^3$ for a time-homogeneous damped tamed 3D Navier-Stokes equation, given by (6.1).

65 Pages, revised version after referee's report

References in corpus (4)

• Global L_2-solutions of stochastic Navier-Stokes equations
• Large deviations for the Boussinesq Equations under Random Influences
• 2D Constrained Navier-Stokes Equations
• Stochastic Constrained Navier-Stokes Equations on ${\mathbb{T}}^2$

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• Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise
• Averaging principle for the stochastic convective Brinkman-Forchheimer equations
• Existence of a martingale solution of the stochastic Hall-MHD equations perturbed by Poisson type random forces on ${\mathbb{R}}^3$
• Martingale solutions of the stochastic Hall-magnetohydrodynamics equations on ${\mathbb{R}}^3$
• Stochastic Navier--Stokes equations on a 3D thin domain