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We consider stationary solutions of the three dimensional Navier--Stokesequations (NS3D) with periodic boundary conditions and driven by an externalforce which might have a deterministic and a random part. The random part ofthe force is white in time and very smooth in space. We investigate smoothnessproperties in space of the stationary solutions. Classical technics forstudying smoothness of stochastic PDEs do not seem to apply since globalexistence of strong solutions is not known. We use the Kolmogorov operator andGalerkin approximations. We first assume that the noise has spatial regularityof order $p$ in the $L^2$ based Sobolev spaces, in other words that its pathsare in $H^p$. Then we prove that at each fixed time the law of the stationarysolutions is supported by $H^{p+1}$. Then, using a totally different technic,we prove that if the noise has Gevrey regularity then at each fixed time, thelaw of a stationary solution is supported by a Gevrey space. Some informationson the Kolmogorov dissipation scale are deduced.

Spatial Smoothness of the Stationary Solutions of the $3D$ Navier-Stokes Equations