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 pge p^->frac{3n}{n+2} (p^->1 in two dimensions). Under additional assumptions we obtain also stochastically strong solutions.
Highlights
Electro-rheological fluids are special smart fluids which change their material properties due to the application of an electric field firstly observed by Winslow [32]
The aim of this paper is to give a rigorous analysis of the following stochastic model for electro-rheological fluids
The first main result of this paper is the existence of a weak martingale solution to (1.5)–(1.6) under periodic boundary conditions where the variable exponent p is Lipschitz continuous in x and satisfies inf see Theorem 2.2 for the precise statement
Summary
Electro-rheological fluids are special smart fluids which change their material properties due to the application of an electric field firstly observed by Winslow [32]. Some (random) derivation from the “real” exponent is expected In this respect, the aim of this paper is to give a rigorous analysis of the following stochastic model for electro-rheological fluids (without loss of generality we assume that = 1 and fe = 0). The first main result of this paper is the existence of a weak martingale solution to (1.5)–(1.6) under periodic boundary conditions where the variable exponent p is Lipschitz continuous in x and satisfies inf. We are concerned with the existence of analytically strong solutions (see Definitions 2.6, 2.12), where Eq (1.5) holds almost everywhere in space This is based on the existence of second derivatives of the velocity field. In the final section we establish the existence of analytically strong solutions subject to suitable additional assumptions imposed on the data
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