Markov selections for the 3D stochastic Navier–Stokes equations

Abstract

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier-Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier-Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.

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

“…We mention only the following one, described carefully in [15]. For other consequences, see [15], [25].…”

“…time. The existence of such a Markov process is not trivial and is dealt with in [15]. A digression on the implications of such a result on the well posedness is given in the final section of the paper.…”

Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations

“…[1,8,9]) and the existence of Markovian selections (cf. [12,13]). We would like to emphasize that the deterministic tamed equation has the following feature: if there exists a bounded solution to the classical 3D Navier-Stokes equation, then this solution also solves the tamed equation.…”

On the martingale problem associated to the $2D$ and $3D$ stochastic Navier–Stokes equations

“…Uniqueness is still an open problem. However, Markov solutions have been constructed and ergodic properties have been proved (see [2]- [4], [7], [9], [11], [12], [22], [25], [26]). In [20], [21], a general form of the stochastic Navier-Stokes equations is derived from the assumptions that the fluid particles are subject to turbulent diffusion.…”

“…However, it has been proved in [4], [7] (see also [11]) that the above assumptions allow constructing a transition semigroup (P t ) t≥0 associated to a Markov family of solutions ((X(t, x)) t≥0 , Ω x , F x , P x ) for x ∈ D(A). Moreover, for sufficiently regular ϕ defined on D(A), P t ϕ is a solution of the Kolmogorov equation associated to (1.2),…”