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.

中文翻译：

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随机影响下的电流变流体：mar和强溶液

我们研究了广义Navier-Stokes方程，该方程控制着随机扰动下的电流变流体的运动。随机效应是通过（i）随机初始数据，（ii）动量方程中由乘性白噪声表示的强迫项和（iii）可变指数\（p = p（\ omega，t， x）\）（由于随机电场的结果）。我们证明了只要变量指数满足\（p \ ge p ^-> \ frac {3n} {n + 2} \）（\（p ^-> 1 \）在二维中），就存在弱mar解决方案。在另外的假设下，我们也获得了随机的强解。