| Abstract: |
| In this talk, we consider the incompressible generalized Navier-Stokes-Voigt equations in a bounded domain $\mathcal{O}\subset\mathbb{R}^d$, $d\geq 2$, driven by a multiplicative Gaussian noise. The considered momentum equation is given by:
$\begin{align*}
\mathrm{d}\left(\boldsymbol{u} - \kappa \Delta \boldsymbol{u}\right) = \left[\boldsymbol{f} +\operatorname{div} \left(-\pi\mathbf{I}+\nu|\mathbf{D}(\boldsymbol{u})|^{p-2}\mathbf{D}(\boldsymbol{u})-\boldsymbol{u}\otimes \boldsymbol{u}\right)\right]\mathrm{d} t + \Phi(\boldsymbol{u})\mathrm{d} \mathrm{W}(t).
\end{align*}$
In the case of $d=2,3$, $\boldsymbol{u}$ accounts for the velocity field, $\pi$ is the pressure, $\boldsymbol{f}$ is a body force and the final term stay for 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 $\mathbf{I}$ for the unit tensor and $\mathbf{D}(\boldsymbol{u}):=\frac{1}{2}\left(\nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^{\top}\right)$ 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 \emph{pathwise uniqueness of solutions}. We employ the classical Yamada-Watanabe theorem to ensure the existence of a unique probabilistic strong solution. This is a joint work with Dr. Ankit Kumar and Dr. H. B. de Oliveira. |
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