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

TL;DR

This paper investigates the existence of solutions to stochastic Navier--Stokes equations modeling electro-rheological fluids with random influences, including initial data, forcing, and variable electric fields, using martingale and pathwise solution frameworks.

Contribution

It establishes the existence of weak martingale solutions and, under further conditions, pathwise solutions for stochastic electro-rheological fluid models with variable exponents.

Findings

Existence of weak martingale solutions under certain exponent conditions.

Pathwise solutions obtained with additional assumptions.

Applicable to fluids with random electric fields and stochastic perturbations.

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 $p\geq p^->\frac{3n}{n+2}$ ($p^->1$ in two dimensions). Under additional assumptions we obtain also pathwise solutions.