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 Oᵈ, d≥ 2, driven by a multiplicative Gaussian noise. The considered momentum equation is given by: {align*} d(u - κ Δ u) = [f +div (-+ν|D(u)|ᵖ⁻²D(u)-u⊗ u)]d t + Φ(u)d W(t). {align*} 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.