Markov selections and their regularity for the three-dimensional stochastic Navier–Stokes equations

Analysis of Equilibrium States of Markov Solutions to the 3D Navier-Stokes Equations Driven by Additive Noise

“…In general one can have several stationary solutions (see for example [20] I and I m ). By the same properties that ensure existence of solutions (and following similar computations, see for example [12]), it is easy to see that I is a compact subset of Ω. Moreover, by Corollary 3.2, I m and hence I e are relatively compact in a much stronger topology.…”

“…In the framework of Markov solutions examined in this paper, the balance of energy corresponds to the a. s. supermartingale property Å¿℄ (and, more generally, of Å ℄) of Definition 2.4. As clarified in [12], the two facts 1. the balance holds only for almost every time, 2. the balance is an inequality, rather than an equality, correspond to a lack of regularity, in time in the first case and in space in the second, of solutions to the equations (1.1). From the point of view of the model, such facts translate to a loss of energy in the balance.…”

“…In this section we recall a few definitions and result from papers [12] and [13], with some additional remarks.…”

“…The weak-strong uniqueness principle. We first extend the weakstrong uniqueness principle stated in [12,Theorem 5.4] to the above problem (A.1). This is the content of the following result.…”

“…In the proof of Theorem 2.6 (see [12]) the sets C (x) are shrunken to one single element in the following way. Fix a family (λ n , f n ) n≥1 which is dense 2 We say that a probability measure P on Ω is associated to a Markov solution (P x ) x∈H if for every t ≥ 0, P| ω B t = P ω(t) for P-a.…”