## 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