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 Electronic Journal of Probability > Vol. 11 (2006) > Paper 27 open journal systems 


Spatial smoothness of the stationary solutions of the 3D Navier--Stokes equations

Cyril Odasso, ENS Cachan, Ker Lann


Abstract
Abstract. We consider stationary solutions of the three dimensional Navier--Stokes equations (NS3D) with periodic boundary conditions and driven by an external force which might have a deterministic and a random part. The random part of the force is white in time and very smooth in space. We investigate smoothness properties in space of the stationary solutions. Classical technics for studying smoothness of stochastic PDEs do not seem to apply since global existence of strong solutions is not known. We use the Kolmogorov operator and Galerkin approximations. We first assume that the noise has spatial regularity of order p in the L2 based Sobolev spaces, in other words that its paths are in Hp. Then we prove that at each fixed time the law of the stationary solutions is supported by Hp+1. Then, using a totally different technic, we prove that if the noise has Gevrey regularity then at each fixed time, the law of a stationary solution is supported by a Gevrey space. Some informations on the Kolmogorov dissipation scale are deduced


Full text: PDF

Pages: 686-699

Published on: August 10, 2006
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Electronic Journal of Probability. ISSN: 1083-6489