Existence and Uniqueness of Weak Solutions for the Generalized Stochastic Navier-Stokes-Voigt Equations

In this work, we consider the incompressible generalized Navier-Stokes-Voigt equations in a bounded domain O⊂Rd, d≥2, driven by a multiplicative Gaussian noise. The considered momentum equation is given by: (Formula presented.) In the case of d=2,3, u accounts for the velocity field, π is the pressure, f is a body force and the final term represents the stochastic forces. Here, κ and ν are given positive constants that account for the kinematic viscosity and relaxation time, and the power-law index p is another constant (assumed p>1) that characterizes the flow. We use the usual notation I for the unit tensor and D(u):=12∇u+(∇u)⊤ for the symmetric part of velocity gradient. For p∈(2dd+2,∞), we first prove the existence of a martingale solution. Then we show the pathwise uniqueness of solutions. We employ the classical Yamada-Watanabe theorem to ensure the existence of a unique probabilistic strong solution.