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

Nathan Glatt-Holtz, Vladimír Šverák, Vlad Vicol

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 languageEnglish (US)
Pages (from-to)619-649
Number of pages31
JournalArchive for Rational Mechanics and Analysis
Volume217
Issue number2
DOIs
StatePublished - 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.
@article{2dcb3cad902d45d097e9c8e2d29771a2,
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.",
author = "Nathan Glatt-Holtz and Vladim{\'i}r Šver{\'a}k and Vlad Vicol",
year = "2015",
month = "8",
day = "10",
doi = "10.1007/s00205-015-0841-6",
language = "English (US)",
volume = "217",
pages = "619--649",
journal = "Archive for Rational Mechanics and Analysis",
issn = "0003-9527",
publisher = "Springer New York",
number = "2",

}

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

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.

UR - http://www.scopus.com/inward/record.url?scp=84930540040&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930540040&partnerID=8YFLogxK

U2 - 10.1007/s00205-015-0841-6

DO - 10.1007/s00205-015-0841-6

M3 - Article

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 -