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

Dominic Breit; Franz Gmeineder

doi:10.1007/s40072-019-00138-6

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

Dominic Breit
, Franz Gmeineder

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

52 Downloads (Pure)

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 language	English
Pages (from-to)	699–745
Number of pages	47
Journal	Stochastics and Partial Differential Equations: Analysis and Computations
Volume	7
Issue number	4
Early online date	18 Mar 2019
DOIs	https://doi.org/10.1007/s40072-019-00138-6
Publication status	Published - Dec 2019

Keywords

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

ASJC Scopus subject areas

Statistics and Probability
Modelling and Simulation
Applied Mathematics

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Cite this

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AU - Breit, Dominic

AU - Gmeineder, Franz

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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

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DO - 10.1007/s40072-019-00138-6

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JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

IS - 4

ER -

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

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Keywords

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