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

Research output: Contribution to journalArticle

### Abstract

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.

Original language English (US) 619-649 31 Archive for Rational Mechanics and Analysis 217 2 https://doi.org/10.1007/s00205-015-0841-6 Published - Aug 10 2015

### Fingerprint

Inviscid Limit
Euler equations
Vorticity
Stochastic Equations
Navier-Stokes Equations
Turbulence
Limiting
Topology
2D Turbulence
Scaling
Drift-diffusion Equations
Weak Topology
Modulus of Continuity
Deterministic Model
Euler Equations
Invariant Measure
Damped
Euler
Model
Iteration

### ASJC Scopus subject areas

• Analysis
• Mathematics (miscellaneous)
• Mechanical Engineering

### Cite this

On Inviscid Limits for the Stochastic Navier–Stokes Equations and Related Models. / Glatt-Holtz, Nathan; Šverák, Vladimír; Vicol, Vlad.

In: Archive for Rational Mechanics and Analysis, Vol. 217, No. 2, 10.08.2015, p. 619-649.

Research output: Contribution to journalArticle

Glatt-Holtz, Nathan ; Šverák, Vladimír ; Vicol, Vlad. / On Inviscid Limits for the Stochastic Navier–Stokes Equations and Related Models. In: Archive for Rational Mechanics and Analysis. 2015 ; Vol. 217, No. 2. pp. 619-649.
title = "On Inviscid Limits for the Stochastic Navier–Stokes Equations and Related Models",
abstract = "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.",
year = "2015",
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