The Martingale Problem for Markov Solutions to the Navier-Stokes Equations

Abstract

Under suitable assumptions of regularity and non-degeneracy on the covariance of the driving additive noise, any Markov solution to the stochastic Navier-Stokes equations has an associated generator of the diffusion and is the unique solution to the corresponding martingale problem. Some elementary examples are discussed to interpret these results.2000 Mathematics Subject Classification. Primary: 76D05; Secondary: 60H15, 35Q30, 60H30, 76M35.

Ergodicity of the 3D stochastic Navier–Stokes equations driven by mildly degenerate noise

“…Existence of at least a Markov solution is ensured by Theorem 3.7 of [14] (see also [12], [15]), for weak martingale solutions that satisfy either a super-martingale type energy inequality ( [14], see also [15] where the authors give an amended version) or an almost sure energy balance ( [24]). More details on the martingale problem associated to these equations can be found in [23].…”

“…Moreover, by Theorem 3.3 of [22], the convergence to the invariant measure is exponentially fast, if the Markov solutions satisfy an almost sure version of the energy inequality (see [22], [24]). The theorem in [22] is proved under an assumption of nondegeneracy of the noise, but the only arguments really used are that the dynamics is strong Feller and irreducible.…”

Critical strong Feller regularity for Markov solutions to the Navier–Stokes equations

“…It is not known whether the martingale problem for the Navier-Stokes equations driven by Gaussian noise is well-posed [7,23]. In order to analyse the problem Da Prato and Debussche [6] (see also [9,18]) showed the existence of Markov processes solutions to the equations and some regularity properties of the transitions semigroups.…”

“…A different approach to the existence and regularity of Markov solutions has been introduced in [13,15] (see also [14,22,23,21,3,25,17]), based on an abstract selection principle for Markov families (see Theorem 2.3) and the short time coupling with a smooth process. A refined analysis of this coupling is one of the purposes of this paper (see Sections 3 and 5.1).…”

“…The main purpose of this paper is to complete the analysis developed in [13,14,15,22,23,21] and relax the regularity assumption, namely to allow coefficients whose decay as |k| → ∞ is of order |σ k | ≈ |k| −3/2−2α 0 for α 0 > 0. In [15] the restriction is α 0 > 1 6 , so the improvement seems tiny.…”