Electro-rheological fluids under random influences: martingale and strong solutions

Dominic Breit, Franz Gmeineder

Research output: Contribution to journalArticle

Abstract

We study generalised Navier–Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent p= p(ω, t, x) (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies p≥p->3nn+2 (p -> 1 in two dimensions). Under additional assumptions we obtain also stochastically strong solutions.

Original languageEnglish
Pages (from-to)699–745
Number of pages47
JournalStochastics and Partial Differential Equations: Analysis and Computations
Volume7
Issue number4
Early online date18 Mar 2019
DOIs
Publication statusE-pub ahead of print - 18 Mar 2019

Fingerprint

Electrorheological fluids
Electrorheological Fluid
Variable Exponent
Strong Solution
Martingale
Stochastic Perturbation
Forcing Term
White noise
Generalized Equation
Random Field
Electric Field
Multiplicative
Momentum
Two Dimensions
Navier-Stokes Equations
Electric fields
Motion
Influence
Character

Keywords

  • Electro-rheological fluids
  • Martingale solution
  • Pathwise solution
  • Stochastic Navier–Stokes equations

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics

Cite this

@article{0f330ab9244b4eb8be41c1c111fde415,
title = "Electro-rheological fluids under random influences: martingale and strong solutions",
abstract = "We study generalised Navier–Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent p= p(ω, t, x) (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies p≥p->3nn+2 (p -> 1 in two dimensions). Under additional assumptions we obtain also stochastically strong solutions.",
keywords = "Electro-rheological fluids, Martingale solution, Pathwise solution, Stochastic Navier–Stokes equations",
author = "Dominic Breit and Franz Gmeineder",
year = "2019",
month = "3",
day = "18",
doi = "10.1007/s40072-019-00138-6",
language = "English",
volume = "7",
pages = "699–745",
journal = "Stochastics and Partial Differential Equations: Analysis and Computations",
issn = "2194-0401",
publisher = "Springer",
number = "4",

}

Electro-rheological fluids under random influences: martingale and strong solutions. / Breit, Dominic; Gmeineder, Franz.

In: Stochastics and Partial Differential Equations: Analysis and Computations, Vol. 7, No. 4, 12.2019, p. 699–745.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Electro-rheological fluids under random influences: martingale and strong solutions

AU - Breit, Dominic

AU - Gmeineder, Franz

PY - 2019/3/18

Y1 - 2019/3/18

N2 - We study generalised Navier–Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent p= p(ω, t, x) (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies p≥p->3nn+2 (p -> 1 in two dimensions). Under additional assumptions we obtain also stochastically strong solutions.

AB - We study generalised Navier–Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent p= p(ω, t, x) (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies p≥p->3nn+2 (p -> 1 in two dimensions). Under additional assumptions we obtain also stochastically strong solutions.

KW - Electro-rheological fluids

KW - Martingale solution

KW - Pathwise solution

KW - Stochastic Navier–Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=85074138842&partnerID=8YFLogxK

U2 - 10.1007/s40072-019-00138-6

DO - 10.1007/s40072-019-00138-6

M3 - Article

VL - 7

SP - 699

EP - 745

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

SN - 2194-0401

IS - 4

ER -