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 $p=p(\omega ,t,x)$ (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies $p\ge p^{-}>\frac{3n}{n+2}$ ($p^{-}>1$ in two dimensions). Under additional assumptions we obtain also pathwise solutions.