Exponential mixing for the $3D$ stochastic Navier--Stokes equations

Cyril Odasso (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We study the Navier-Stokes equations in dimension $3$ (NS3D) driven by a noise which is white in time. We establish that if the noise is at same time sufficiently smooth and non degenerate in space, then the weak solutions converge exponentially fast to equilibrium. We use a coupling method. The arguments used in dimension two do not apply since, as is well known, uniqueness is an open problem for NS3D. New ideas are introduced. Note however that many simplifications appears since we work with non degenerate noises.

Mots clés: Stochastic three-dimensional Navier-Stokes equations; Markov transition semi-group; invariant measure; ergodicity; coupling method; exponential mixing; galerkin approximation

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Cyril Odasso. Exponential mixing for the $3D$ stochastic Navier--Stokes equations. Communications in Mathematical Physics, 2007, 270 (1), pp.109-139. ⟨10.1007/s00220-006-0156-4⟩. ⟨hal-00015078v2⟩