Appl Math Optim 53:141–161 (2006)
2005 Springer Science+Business Media, Inc.
The Asymptotic Behaviour of a Stochastic 3D LANS-α Model
Tom ´as Caraballo, Antonio M. M´arquez-Dur´an, and Jos´e Real
Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla,
Apdo. Correos 1160, 41080-Sevilla, Spain
Abstract. The long-time behaviour of a stochastic 3D LANS-α model on a bounded
domain is analysed. First, we reformulate the model as an abstract problem. Next,
we establish sufﬁcient conditions ensuring the existence of stationary (steady state)
solutions of this abstract nonlinear stochastic evolution equation, and study the sta-
bility properties of the model. Finally, we analyse the effects produced by stochas-
tic perturbations in the deterministic version of the system (persistence of expo-
nential stability as well as possible stabilisation effects produced by the noise).
The general results are applied to our stochastic LANS-α system throughout the
Key Words. LANS-α equations, Cylindrical Wiener process, Asymptotic be-
AMS Classiﬁcation. 60H15, 35Q35, 60H30, 35R15.
In this paper we are mainly interested in the study of the asymptotic behaviour of solutions
of the 3D Lagrangian averaged Navier–Stokes (LANS-α) equations, with homogeneous
Dirichlet boundary conditions in a bounded domain, in the case in which random per-
turbations appear. To be more precise, let D be a connected and bounded open subset
, with a C
boundary ∂ D. We denote by A the Stokes operator, and consider the
This research was partly supported by Ministerio de Ciencia y Tecnolog´ıa (Spain) and FEDER (Euro-
pean Community) Grant BFM2002-03068, and Junta de Andaluc´ıa Project FQM314.