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Characterization of the law for 3D stochastic hyperviscous fluids. (English) Zbl 1336.76028
Summary: We consider the 3D hyperviscous Navier-Stokes equations in vorticity form, where the dissipative term \(-\Delta \vec \xi \) of the Navier-Stokes equations is substituted by \((-\Delta )^{1+c} \vec \xi \). We investigate how big the correction term \(c\) has to be in order to prove, by means of Girsanov transform, that the vorticity equations are equivalent (in law) to easier reference equations obtained by neglecting the stretching term. This holds as soon as \(c>\frac 12\), improving previous results obtained with \(c>\frac 32\) in a different setting in [the author, Commun. Stoch. Anal. 2, No. 2, 209–227 (2008; Zbl 1331.35407); erratum ibid. 5, No. 2, 431–432 (2011); S. M. Kozlov, Tr. Semin. Im. I. G. Petrovskogo 4, 147–172 (1978; Zbl 0416.60074)].

MSC:
76M35 Stochastic analysis applied to problems in fluid mechanics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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