Dynamics of stochastic 2D Navier-Stokes equations
type: Item
instance of: scholarly article
title: Dynamics of stochastic 2D Navier-Stokes equations
zbMATH Open document ID: 1213.60107
published in: Journal of Functional Analysis
publication date: 2010-05-17T00:00:00Z
review text: The authors study mainly two aspects of solutions of 2\(D\) stochastic Navier-Stokes equations driven by a multiplicative white noise with Dirichlet boundary conditions, namely existence of a perfect locally compacting \(C^{1,1}\) cocycle generated by all solutions of the equation, and the characterization of the local behavior of solutions near an equilibrium or a stationary point. The results are presented in Sections 3 and 4. In Section 3, the authors address the first problem and give in Theorem 3.2 the main result of this section i.e., that strong solutions of the stochastic Navier-Stokes equation generates a Fréchet \(C^{1,1}\) locally compacting cocycle. This result is obtained by combining a variational technique which allows one to transform the equation into a random Navier-Stokes equation and by using Galerkin approximations and a priori estimates. Section 4 is devoted to the study of the local behavior of solutions near an equilibrium or near a stationary point. First, the later notion is defined in Definition 4.1 and then, in section 4.1, the dynamics near a general equilibrium are described using a countable non-random Lyapunov spectrum of the associated Osledec operator (see Theorem 4.1 and Theorem 4.2 for hyperbolic stationary random solutions). Finally Section 4.2 is devoted to the same study but when the equilibrium is identically zero. Among other results, the authors provide the Lyapunov spectrum (see Theorem 4.3), as well as sufficient conditions on the parameters of the coefficients and on the geometry of the domain to obtain that zero is the unique equilibrium (see Theorem 4.6), and so-called local and global invariant manifolds Theorems for the stochastic Navier-Stokes equation (we refer to Theorem 4.6).
reviewed by: Anthony Réveillac
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 37L55
zbMATH DE Number: 5708618
zbMATH Keywords: stochastic Navier-Stokes equation
zbMATH Keywords: Cocycle
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: Stable manifolds
zbMATH Keywords: Invariant manifolds
author: Salah Mohammed
author: Tu-Sheng Zhang
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.11.007
OpenAlex ID: W183939149
cites work: Stochastic Navier-stokes equations with multiplicative noise
cites work: Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains
cites work: Exponentially stable stationary solutions for stochastic evolution equations and their perturba\-tion
cites work: Stochastic Equations in Infinite Dimensions
cites work: Ergodicity for Infinite Dimensional Systems
cites work: Invariant manifolds for stochastic partial differential equations.
cites work: Smooth stable and unstable manifolds for stochastic evolutionary equations
cites work: Q4850166
cites work: Stochastic differential equations in fluid dynamics
cites work: A large deviation principle for 2D stochastic Navier-Stokes equation
cites work: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing
cites work: Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs
cites work: The stable manifold theorem for non-linear stochastic systems with memory. II: The local stable manifold theorem.
cites work: The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations
cites work: Introduction (to ``From finite to infinite dimensional dynamical systems'')
cites work: Characteristic exponents and invariant manifolds in Hilbert space
cites work: Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise
cites work: Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations
cites work: Infinite-dimensional dynamical systems in mechanics and physics
DOI: 10.1016/J.JFA.2009.11.007
Recommended article: Stochastic 2-D Navier-Stokes equation
Recommended article: Stochastic two-dimensional Navier-Stokes equations on time-dependent domains
Recommended article: 2D stochastic Navier-Stokes equations driven by jump noise
Recommended article: Stochastic Navier-Stokes equations
Recommended article: Q2978452
Recommended article: Stochastic Navier-Stokes equations and related models
Recommended article: Q3297902
Recommended article: Q4659178
description: scientific article; zbMATH DE number 5708618