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  and .
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 and stochastic Navier–Stokes equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19 (2008), no. 3, pp. 247–264DOI 10.4171/RLM/523