Stochastic Navier-Stokes equations: Analysis of the noise to have a unique invariant measure

In [6], it has been proven that there exists a unique invariant measure for the 2D Navier-Stokes equations perturbed by a white noise term; this is the probability measure representing the asymptotic behavior. There, the assumptions on the noise were quite restrictive. In this paper we remove the heaviest limitation, that is the lower bound on the range of the noise covariance, providing a complete analysis of sufficient conditions for the existence of a unique invariant measure.

[1]  Yoshikazu Giga,et al.  Solutions in Lr of the Navier-Stokes initial value problem , 1985 .

[2]  Jerzy Zabczyk,et al.  Strong Feller Property and Irreducibility for Diffusions on Hilbert Spaces , 1995 .

[3]  Franco Flandoli,et al.  Ergodicity of the 2-D Navier-Stokes equation under random perturbations , 1995 .

[4]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[5]  Jan Seidler,et al.  Ergodic behaviour of stochastic parabolic equations , 1997 .

[6]  Franco Flandoli,et al.  Stochastic differential equations in fluid dynamics , 1996 .

[7]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[8]  Benedetta Ferrario,et al.  Ergodic results for stochastic navier-stokes equation , 1997 .

[9]  R. Khas'minskii.  Ergodic Properties of Recurrent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations , 1960 .

[10]  J. Doob.  Asymptotic properties of Markoff transition prababilities , 1948 .

[11]  J. Zabczyk,et al.  Strong feller property for stochastic semilinear equations , 1995 .

[12]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[13]  Franco Flandoli,et al.  Dissipativity and invariant measures for stochastic Navier-Stokes equations , 1994 .

[14]  Jan Seidler,et al.  Ergodic properties of recurrent solutions of stochastic evolution equations , 1994 .

[15]  K. Elworthy,et al.  Formulae for the Derivatives of Heat Semigroups , 1994, 1911.10971.