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Martingale and stationary solutions for stochastic non-Newtonian fluids. (English) Zbl 1240.60184

A stochastic isothermal nonlinear incompressible bipolar non-Newtonian equation \( du+(u\cdot \nabla u-\nabla \cdot \tau (e(u))+\nabla \pi )\, dt=f(x)\,dt+G(u)\,dW \) for a fluid in a bounded domain \(D\subseteq \mathbb {R}^n\) (\(n\in \{2,3\}\)) is considered with a no-slip boundary condition and, it is also imposed that the first moments of the traction vanish on \(\partial D\). Here, \(u\) represents the velocity of the fluid, \(f\) is a deterministic forcing term, \(\pi \) is the pressure, \(G\) is continuous of at most linear growth, \(W\) is a cylindrical Wiener process and \(\tau (e(u))\) is a symmetric stress tensor defined by \( \tau (e(u))=2\mu _0(\varepsilon +| e(u)| ^2)^\frac {p-2}{2}e(u)-2\mu _1\Delta e(u),\quad e(u)=\frac 12\left (\partial _ju_i+\partial _iu_j\right ) \), where \(\mu _0>0\), \(\mu _1>0\), \(\varepsilon >0\) and \(p\in (1,\infty )\).
A martingale solution is proved to exist provided that \(p\in (1,2)\cup (2,2+\frac {2}{2+n}]\). Moreover, if \(\mu _1\) is sufficiently large, then a stationary martingale solution exists.
As far as the proof is concerned, a sequence of solutions of approximating finite-dimensional problems is considered and shown to be tight in a suitable space (by a compactness argument), and then the martingale solutions are constructed using the Skorokhod representation theorem and a representation theorem for martingales.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
60G10 Stationary stochastic processes
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