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On Inviscid Limits for the Stochastic Navier–Stokes Equations and Related Models

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We study inviscid limits of invariant measures for the 2D stochastic Navier–Stokes equations. As shown by Kuksin (J Stat Phys 115(1–2):469–492, 2004), the noise scaling \({\sqrt{\nu}}\) is the only one which leads to non-trivial limiting measures, which are invariant for the 2D Euler equations. Using a Moser-type iteration for stochastic drift-diffusion equations, we show that any limiting measure \({\mu_{0}}\) is in fact supported on bounded vorticities. Relationships of \({\mu_{0}}\) to the long term dynamics of 2D Euler in \({L^{\infty}}\) with the weak* topology are discussed. We also obtain a drift-independent modulus of continuity for a stationary deterministic model problem, which leads us to conjecture that in fact \({\mu_0}\) is supported on \({C^0}\) . Moreover, in view of the Batchelor–Krainchnan 2D turbulence theory, we consider inviscid limits for a weakly damped stochastic Navier–Stokes equation. In this setting we show that only an order zero noise scaling (with respect to ν) leads to a nontrivial limiting measure in the inviscid limit.

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Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Nathan Glatt-Holtz

  2. Department of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Vladimír Šverák

  3. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Vlad Vicol

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  1. Nathan Glatt-Holtz
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  2. Vladimír Šverák
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Glatt-Holtz, N., Šverák, V. & Vicol, V. On Inviscid Limits for the Stochastic Navier–Stokes Equations and Related Models. Arch Rational Mech Anal 217, 619–649 (2015). https://doi.org/10.1007/s00205-015-0841-6

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