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

Abstract

In this work, we consider the incompressible generalized Navier-Stokes-Voigt equations in a bounded domain , , driven by a multiplicative Gaussian noise. The considered momentum equation is given by:

In the case of , accounts for the velocity field, is the pressure, is a body force and the final term represents 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 ) that characterizes the flow. We use the usual notation for the unit tensor and for the symmetric part of velocity gradient. For , 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.