Electro-rheological fluids under random influences: martingale and strong solutions
Dominic Breit, Franz Gmeineder · Stochastic Partial Differential Equations Analysis and Computations · 2019
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 stochastically strong solutions.