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

Dominic Breit, Franz Gmeineder

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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 statusPublished - 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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