JournalsrlmVol. 19 , No. 3DOI 10.4171/rlm/523

On the martingale problem associated to the 2<em>D</em> and 3<em>D</em> stochastic Navier–Stokes equations

  • Giuseppe Da Prato

    Scuola Normale Superiore, Pisa, Italy
  • Arnaud Debussche

    Antenne de Bretagne, Bruz, France
On the martingale problem associated to the 2<em>D</em> and 3<em>D</em> stochastic Navier–Stokes equations cover

Abstract

In this paper we consider a Markov semigroup (Pt)t0(P_t)_{t\ge 0} associated to 2D2D and 3D3D Navier-Stokes equations. In the two-dimensional case PtP_t is unique, whereas in the three-dimensional case (where uniqueness is not known) it is constructed as in \cite{DPD-NS3D} and \cite{DO06}. For d=2d=2, we explicit a core, identify the abstract generator of (Pt)t0(P_t)_{t\ge 0} with the differential Kolmogorov operator LL on this core and prove existence and uniqueness for the corresponding martingale problem. In dimension 33, 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 (St)t0,(S_t)_{t\ge 0}, obtained by adding a suitable potential and then using the Feynman--Kac formula. Then we identify the abstract generator (St)t0(S_t)_{t\ge 0} with a differential operator NN on this core and prove uniqueness for the stopped martingale problem.