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, ItalyArnaud Debussche
Antenne de Bretagne, Bruz, France

Abstract
In this paper we consider a Markov semigroup associated to and Navier-Stokes equations. In the two-dimensional case 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 , we explicit a core, identify the abstract generator of with the differential Kolmogorov operator on this core and prove existence and uniqueness for the corresponding martingale problem. In dimension , 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 obtained by adding a suitable potential and then using the Feynman--Kac formula. Then we identify the abstract generator with a differential operator on this core and prove uniqueness for the stopped martingale problem.
Cite this article
Giuseppe Da Prato, Arnaud Debussche, On the martingale problem associated to the 2<em>D</em> and 3<em>D</em> stochastic Navier–Stokes equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19 (2008), no. 3, pp. 247–264
DOI 10.4171/RLM/523