Global-in-time probabilistically strong solutions to stochastic power-law equations: Existence and non-uniqueness

Abstract

We are concerned with the power-law fluids driven by an additive stochastic forcing in dimension d⩾3. For the power index r∈(1,3d+2d+2), we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in Llocp([0,∞);L2)∩C([0,∞);W1,max{1,r−1}),p⩾1 for every divergence free initial condition in L2∩W1,max{1,r−1}. This result in particular implies non-uniqueness in law. Our result is sharp in the three dimensional case in the sense that the solution is unique if r⩾3d+2d+2.