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⊂ℝ^d, d≥ 2, driven by a multiplicative Gaussian noise. The considered momentum equation is given by: d(u - κΔu) = [f +div(-π𝐈+ν|𝐃(u)|^p-2𝐃(u)-u⊗u)]d t + Φ(u)dW(t). 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 stay for 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):=1/2(∇u + (∇u)^⊤) for the symmetric part of velocity gradient. For p∈(2d/d+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.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.