On the method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations

Abstract

3D stochastic Navier-Stokes equations with a suitable nondegenerate noise are considered. Following a method introduced by Da Prato and Debussche, it is proved that every Markov process associated to the equations has a Strong Feller like continuity property with respect to initial conditions.

Markov selections for the 3D stochastic Navier–Stokes equations

“…The proof in [6] is based on a completely different argument, longer but at present stronger from the viewpoint of quantitative estimates on derivatives of the Kolmogorov semi-group. In Flandoli [13] one can see a variation on the proof of [6] applied to the Markov selections constructed here.…”

“…for every ε > 0 (see [13]), and in particular z ∈ C([0, ∞); W ), since θ(α 0 ) < α 0 + 1 2 . Given ω ∈ Ω * NS , one can prove, by standard arguments (see for example [17]) that Eq.…”

“…Lemma C.3 (Continuity along the controllers) Let s, p and α 1 be chosen as above, and let w n → w in W s, p ([0, T ]; D(A α 1 )). Let u be the solution to equation (C. 13) corresponding to w and some initial condition x and let τ = inf{t ≥ 0 : |u(t)| 2 W ≥ R} (and τ = +∞ if the set is empty). Define similarly, for each n ∈ N, u n and τ n corresponding to w n and the same initial condition x.…”

Stochastic Three-Dimensional Rotating Navier–Stokes Equations: Averaging, Convergence and Regularity

“…The symbols of the operators S and curl are related by (21). Clearly, Q = Q if the operators Q and S commute (equivalently, Q and curl commute).…”

“…Additional information on SPDEs can be found in [8,16,17,21,24,25]. Our choice of the space H 3 in the above definition is natural for considering solutions of the stochastic three-dimensional NSEs in the inviscid limit ν → 0 (under appropriate scalings).…”