Strong solution for a stochastic model of two-dimensional second grade fluids: Existence, uniqueness and asymptotic behavior

Abstract

We investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of R 2 . Global existence and uniqueness of strong probabilistic solution is established. In contrast to previous results on this model we show that the sequence of Galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem. We also give two results on the long time behaviour of the solution. Mainly we prove that the strong solution of our stochastic model converges exponentially in mean square to the stationary solution of the time-independent second grade fluids equations if the deterministic part of the external force does not depend on time. If the deterministic forcing term explicitly depends on time, then the strong probabilistic solution decays exponentially in mean square.

Strong solutions for the stochastic 3D LANS-α model driven by non-Gaussian Lévy noise

“…In particular, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solutions (see Theorem 3). The proof follows the approach in [13,14,45]. This article is therefore a generalization of the papers [11,13] to the case of nonGaussian Lévy noise.…”

Optimal control for two-dimensional stochastic second grade fluids

“…To study the well-posedness of the state equations (1.1), we should impose a suitable boundary condition on the boundary Γ of the domain O. Considering the classical homogeneous Dirichlet boundary condition, the authors in [26] proved the existence and uniqueness of strong stochastic solutions. Other physically relevant boundary condition is the so-called Navier-slip boundary condition, which reads as Y · n = 0, (n · DY ) · τ = 0 on Γ, (1.2) where n = (n 1 , n 2 ) and τ = (−n 2 , n 1 ) are the unit normal and tangent vectors, respectively, to the boundary Γ and DY = 1 2 ∇Y + ∇Y ⊤ is the rate-of-strain tensor.…”

Anticipating stochastic equation of two-dimensional second grade fluids

“…They also have interesting connections with other fluid models, see [1,2,3]. For researchs on stochastic models of 2D second grade fluids, we refer to [12,13,15,17,16,14].…”