THE KOLMOGOROV EQUATION ASSOCIATED TO THE STOCHASTIC NAVIER–STOKES EQUATIONS IN 2D

2004 ◽  
Vol 07 (02) ◽  
pp. 163-182 ◽  
Author(s):  
VIOREL BARBU ◽  
GIUSEPPE DA PRATO ◽  
ARNAUD DEBUSSCHE
Keyword(s):  
Invariant Measure ◽  
Stokes Equations ◽  
Dirichlet Boundary Conditions ◽  
Kolmogorov Equation ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Sharp Estimates ◽  
Stochastic Term ◽  
Derivatives Of

We consider a 2D Navier–Stokes equation with Dirichlet boundary conditions perturbed by a stochastic term of the form [Formula: see text], where Q is a non-negative operator and Ẇ is a spacetime white noise. We solve the corresponding Kolmogorov equation in the space L2(H,ν) where ν is an invariant measure and prove the "carré du champs" identity. The key tool are some sharp estimates on the derivatives of the solution. Our main assumption is that the viscosity is large compared with the norm of Q. As a byproduct, we give a new simple proof of the uniqueness of the invariant measure and obtain an existence result for a Hamilton–Jacobi equation related to a control problem.

Large-frequency global regularity for the incompressible Navier–Stokes equation

1999 ◽  
Vol 129 (5) ◽  
pp. 903-912
Author(s):  
Joel D. Avrin
Keyword(s):  
Boundary Conditions ◽  
Global Regularity ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Thin Domains ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Incompressible Navier Stokes Equation

We obtain global existence and regularity of strong solutions to the incompressible Navier–Stokes equations for a variety of boundary conditions in such a way that the initial and forcing data can be large in the high-frequency eigenspaces of the Stokes operator. We do not require that the domain be thin as in previous analyses. But in the case of thin domains (and zero Dirichlet boundary conditions) our results represent a further improvement and refinement of previous results obtained.

scholarly journals Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

2020 ◽  
Vol 22 (2) ◽  
Author(s):  
Zdzisław Brzeźniak ◽  
Gaurav Dhariwal
Keyword(s):  
Invariant Measure ◽  
Stokes Equation ◽  
Diffusion Processes ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Uniqueness Of Solutions ◽  
Unique Strong Solution ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Whole Space

Abstract Röckner and Zhang (Probab Theory Relat Fields 145, 211–267, 2009) 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 Stroock and Varadhan (Multidimensional diffusion processes, Springer, Berlin, 1979). 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 $$L^4$$ L 4 -norm of the solution from the torus to $${\mathbb {R}}^3$$ R 3 , see Lemma 5.1 and thus establish the existence of an invariant measure on $${\mathbb {R}}^3$$ R 3 for a time-homogeneous damped tamed 3D Navier–Stokes equation, given by (6.1).

scholarly journals Hyperviscous stochastic Navier–Stokes equations with white noise invariant measure

2020 ◽  
Vol 20 (06) ◽  
pp. 2040005
Author(s):  
M. Gubinelli ◽  
M. Turra
Keyword(s):  
Invariant Measure ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Absolutely Continuous ◽  
Navier Stokes Equations ◽  
Whole Space ◽  
Martingale Solutions

We prove existence and uniqueness of martingale solutions to a (slightly) hyper-viscous stochastic Navier–Stokes equation in 2d with initial conditions absolutely continuous with respect to the Gibbs measure associated to the energy, getting the results both in the torus and in the whole space setting.

scholarly journals Generalized Navier–Stokes Equations and Dynamics of Plane Molecular Media

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 288
Author(s):  
Alexei Kushner ◽  
Valentin Lychagin
Keyword(s):  
Carbon Dioxide ◽  
Internal Structure ◽  
Stokes Equation ◽  
Hydrogen Cyanide ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

The first analysis of media with internal structure were done by the Cosserat brothers. Birkhoff noted that the classical Navier–Stokes equation does not fully describe the motion of water. In this article, we propose an approach to the dynamics of media formed by chiral, planar and rigid molecules and propose some kind of Navier–Stokes equations for their description. Examples of such media are water, ozone, carbon dioxide and hydrogen cyanide.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

The Onset of Turbulence: Insights Into the Navier-Stokes Equations

2017 ◽  
Author(s):  
Carl E. Rathmann
Keyword(s):  
Stokes Equations ◽  
Direct Consequence ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Fluid Behavior ◽  
Onset Of Turbulence ◽  
And Mathematics ◽  
High Reynolds

For well over 150 years now, theoreticians and practitioners have been developing and teaching students easily visualized models of fluid behavior that distinguish between the laminar and turbulent fluid regimes. Because of an emphasis on applications, perhaps insufficient attention has been paid to actually understanding the mechanisms by which fluids transition between these regimes. Summarized in this paper is the product of four decades of research into the sources of these mechanisms, at least one of which is a direct consequence of the non-linear terms of the Navier-Stokes equation. A scheme utilizing chaotic dynamic effects that become dominant only for sufficiently high Reynolds numbers is explored. This paper is designed to be of interest to faculty in the engineering, chemistry, physics, biology and mathematics disciplines as well as to practitioners in these and related applications.

Navier Stokes Derivative Estimates in Three Dimensions with Boundary Values and Body Forces

1991 ◽  
Vol 43 (6) ◽  
pp. 1161-1212 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Stokes Equations ◽  
Finite Energy ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Boundary Values ◽  
Vector Solution ◽  
Navier Stokes Equations ◽  
Body Forces ◽  
Derivatives Of

AbstractFor a vector solution u(x, t) with finite energy of the Navier Stokes equations with body forces and boundary values on a region Ω ⊆ R3 for t > 0, conditions are established on the L6/5(Ω) and L2(Ω) norms of derivatives of the data that ensure the estimates and max , up to any given integer value of the weighted order 2r+s, where r or s = s1 + s2 + s3 > 0 and 0 < T < ∞.

ON THE ASYMPTOTIC ANALYSIS OF THE BGK MODEL TOWARD THE INCOMPRESSIBLE LINEAR NAVIER–STOKES EQUATION

2010 ◽  
Vol 20 (08) ◽  
pp. 1299-1318 ◽  
Author(s):  
A. BELLOUQUID
Keyword(s):  
Asymptotic Analysis ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Uniqueness Theorems ◽  
Bgk Model ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Stokes Systems ◽  
Global Strong Solutions

This paper deals with the analysis of the asymptotic limit for BGK model to the linearized Navier–Stokes equations when the Knudsen number ε tends to zero. The uniform (in ε) existence of global strong solutions and uniqueness theorems are proved for regular initial fluctuations. As ε tends to zero, the solution of BGK model converges strongly to the solution of the linearized Navier–Stokes systems. The validity of the BGK model is critically analyzed.

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