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

Abstract

In this paper we consider a Markov semigroup $(P_t{)}_{t\ge 0}$ associated to $2D$ and $3D$ Navier–Stokes equations. In the two-dimensional case $P_t$ is unique, whereas in the three-dimensional case (where uniqueness is not known) it is constructed as in [4] and [7].

For $d=2$, we explicit a core, identify the abstract generator of $(P_t{)}_{t\ge 0}$ with the differential Kolmogorov operator $L$ on this core and prove existence and uniqueness for the corresponding martingale problem. In dimension $3$, we are not able to prove a similar result and we explain the difficulties encountered. Nonetheless, we explicit a core for the generator of the transformed semigroup $(S_t{)}_{t\ge 0},$ obtained by adding a suitable potential and then using the Feynman–Kac formula. Then we identify the abstract generator $(S_t{)}_{t\ge 0}$ with a differential operator $N$ on this core and prove uniqueness for the stopped martingale problem.