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{align*} \mathrm{d}\left(\boldsymbol{u} - κΔ\boldsymbol{u}\right) = \left[\boldsymbol{f} +\operatorname{div} \left(-π\mathbf{I}+ν|\mathbf{D}(\boldsymbol{u})|^{p-2}\mathbf{D}(\boldsymbol{u})-\boldsymbol{u}\otimes \boldsymbol{u}\right)\right]\mathrm{d} t + Φ(\boldsymbol{u})\mathrm{d} \mathrm{W}(t). \end{align*} In the case of $d=2,3$, $\mathbf{u}$ accounts for the velocity field, $\ddot{I}€$ is the pressure, $\mathbf{f}$ is a body force and the final term stay for the stochastic forces. Here, $\hat{I}º$ and $\hat{I}½$ 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 $\mathbf{I}$ for the unit tensor and $\mathbf{D}(\mathbf{u}):=\frac{1}{2}\left(\nabla \mathbf{u}+(\nabla \mathbf{u}{)}^{\top }\right)$ for the symmetric part of velocity gradient. For $p\in (\frac{2d}{d+2},\infty )$, 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.