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 - u) = +div (-+|D (u) |^p-2D (u) -u u) t + (u) d W (t). align* In the case of d=2, 3, u accounts for the velocity field, is the pressure, f is a body force and the final term stay for the stochastic forces. Here, and 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. 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.