TY - JOUR

T1 - On Inviscid Limits for the Stochastic Navier–Stokes Equations and Related Models

AU - Glatt-Holtz, Nathan

AU - Šverák, Vladimír

AU - Vicol, Vlad

N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2015/8/10

Y1 - 2015/8/10

N2 - 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 $${\sqrt{\nu}}$$ν 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 $${\mu_{0}}$$μ0 is in fact supported on bounded vorticities. Relationships of $${\mu_{0}}$$μ0 to the long term dynamics of 2D Euler in $${L^{\infty}}$$L∞ 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 $${\mu_0}$$μ0 is supported on $${C^0}$$C0. 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.

AB - 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 $${\sqrt{\nu}}$$ν 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 $${\mu_{0}}$$μ0 is in fact supported on bounded vorticities. Relationships of $${\mu_{0}}$$μ0 to the long term dynamics of 2D Euler in $${L^{\infty}}$$L∞ 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 $${\mu_0}$$μ0 is supported on $${C^0}$$C0. 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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U2 - 10.1007/s00205-015-0841-6

DO - 10.1007/s00205-015-0841-6

M3 - Article

AN - SCOPUS:84930540040

VL - 217

SP - 619

EP - 649

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -