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Exponential Mixing for the 3D Stochastic Navier–Stokes Equations

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Abstract

We study the Navier–Stokes equations in dimension 3 (NS3D) driven by a noise which is white in time. We establish that if the noise is at the same time sufficiently smooth and non-degenerate in space, then the weak solutions converge exponentially fast to equilibrium.

We use a coupling method. The arguments used in dimension two do not apply since, as is well known, uniqueness is an open problem for NS3D. New ideas are introduced. Note however that many simplifications appear since we work with non degeneratenoises.

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  1. Ecole Normale Supérieure de Cachan, Antenne de Bretagne, Avenue Robert Schuman, Campus de Ker Lann, 35170, Bruz, France

    Cyril Odasso

  2. IRMAR, UMR 6625 du CNRS, Campus de Beaulieu, 35042, Rennes Cedex, France

    Cyril Odasso

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  1. Cyril Odasso
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Communicated by G. Gallavotti

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Odasso, C. Exponential Mixing for the 3D Stochastic Navier–Stokes Equations. Commun. Math. Phys. 270, 109–139 (2007). https://doi.org/10.1007/s00220-006-0156-4

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  • DOI: https://doi.org/10.1007/s00220-006-0156-4

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