Existence theory for stochastic power law fluids

We consider the equations of motion for an incompressible non-Newtonian fluid in a bounded Lipschitz domain $${G {\subset} \mathbb{R}^{d}}$$G⊂Rd during the time interval (0, T) together with a stochastic perturbation driven by a Brownian motion W. The balance of momentum reads as$${\rm {d}\mathbf{v}} = {\rm div}\mathbf{S} {\rm d}t-(\nabla \mathbf{v})\mathbf{v} {\rm d}t + \nabla\pi {\rm d}t + \mathbf{f}{\rm d}t + \Phi(\mathbf{v}) {\rm d}\mathbf{W}_{t},$$dv=divSdt-(∇v)vdt+∇πdt+fdt+Φ(v)dWt,where v is the velocity, $${\pi}$$π the pressure and f an external volume force. We assume the common power law model $${\mathbf{S}(\varepsilon(\mathbf{v}))=(1+|\varepsilon(\mathbf{v})|)^{p-2}\varepsilon(\mathbf{v})}$$S(ε(v))=(1+|ε(v)|)p-2ε(v) and show the existence of martingale weak solution provided $${p > \frac{2d+2}{d+2}}$$p>2d+2d+2. Our approach is based on the $${L^{\infty}}$$L∞-truncation and a harmonic pressure decomposition which are adapted to the stochastic setting.