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 \(\mathcal {O}\subset \mathbb {R}^d\), \(d\ge 2\), driven by a multiplicative Gaussian noise. The considered momentum equation is given by:

$$\begin{aligned} \textrm{d}\left( \varvec{u} - \kappa \Delta \varvec{u}\right) = \left[ \varvec{f} +{\operatorname {div}} \left( -\pi \textbf{I}+\nu |\textbf{D}(\varvec{u})|^{p-2}\textbf{D}(\varvec{u})-\varvec{u}\otimes \varvec{u}\right) \right] \textrm{d} t + \Phi (\varvec{u})\textrm{dW}(t). \end{aligned}$$

In the case of \(d=2,3\), \(\varvec{u}\) accounts for the velocity field, \(\pi \) is the pressure, \(\varvec{f}\) is a body force and the final term represents the stochastic forces. Here, \(\kappa \) and \(\nu \) 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 \(\textbf{I}\) for the unit tensor and \(\textbf{D}(\varvec{u}):=\frac{1}{2}\left( \nabla \varvec{u} + (\nabla \varvec{u})^{\top }\right) \) for the symmetric part of velocity gradient. For \(p\in \big (\frac{2d}{d+2},\infty \big )\), 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.