Martingale solutions and Markov selections for stochastic partial differential equations

Three-dimensional Navier–Stokes equations driven by space–time white noise

“…In the two dimensional case existence and uniqueness of strong solutions have been obtained if the noisy forcing term is white in time and colored in space. In the three dimensional case, existence of martingale (=probabilistic weak) solutions, which form a Markov selection, have been constructed for the stochastic 3D Navier-Stokes equation driven by trace-class noise in [10], [7], [13]. Furthermore, the ergodicity has been obtained for every Markov selection of the martingale solutions if driven by non-degenerate trace-class noise (see [10]).…”

Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations

“…where Φ is given by (13). Thanks to h ∈ A T M , i.e., T 0 h (s, ω) 2 l 2 ds M , by Girsanov's theorem, u solves the following control equation:…”

“…Note that the law of h is just ν(·, C T (U)). By Skorohod's representation 13 theorem, there are random variablesh k ,W k , k ∈ N andh,W on a probability space (Ω,P ) such that (1) (h k ,W k ) a.s. converges to (h,W ) in D N × C T (U); (2) (h k ,W k ) has the same law as (h k , W ) for each k ∈ N; (3) The law of {h,W } is ν, and the law of h is the same ash. We remark that in the proof of Lemma 3.3, one can replace (h k , W ) by (h k ,W k ).…”

“…Without the term g N , equation (1) is the stochastic 3D Navier-Stokes equation for which it is well known that there exists a martingale solution (cf. [15,13]). But the uniqueness is open, and one only knows the existence of a locally unique strong solution (cf.…”

“…[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.…”