Existence and nonuniqueness of ergodic Leray–Hopf solutions to the stochastic power-law flows

We study long time behavior of shear-thinning fluid flows in d≥3 dimensions, driven by additive stochastic forcing of trace class, with power-law indices ranging from 1 to 2dd+2. We particularly focus on Leray–Hopf solutions, that is, on analytically weak solutions satisfying energy inequality. Introducing a new kind of energy related functional into the technique of convex integration enables the construction of infinitely many such solutions that are probabilistically strong for a certain initial value. Furthermore, we provide global in time estimates which lead to the existence of infinitely many stationary and even ergodic Leray–Hopf solutions. These results represent the first construction of Leray–Hopf solutions in the framework of stochastic shear-thinning fluids within this range of power-law indices.