Existence and uniqueness of weak solutions for the generalized stochastic Navier-Stokes-Voigt equations
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2024
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| author | Kumar, Ankit de Oliveira, Hermenegildo Borges Mohan, Manil T. |
| author_facet | Kumar, Ankit de Oliveira, Hermenegildo Borges Mohan, Manil T. |
| contents | In this work, 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} - 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 $d=2,3$, $\boldsymbol{u}$ accounts for the velocity field, $π$ is the pressure, $\boldsymbol{f}$ is a body force and the final term stay for the stochastic forces. Here, $κ$ and $ν$ 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08001 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and uniqueness of weak solutions for the generalized stochastic Navier-Stokes-Voigt equations Kumar, Ankit de Oliveira, Hermenegildo Borges Mohan, Manil T. Probability In this work, 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} - 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 $d=2,3$, $\boldsymbol{u}$ accounts for the velocity field, $π$ is the pressure, $\boldsymbol{f}$ is a body force and the final term stay for the stochastic forces. Here, $κ$ and $ν$ 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 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. |
| title | Existence and uniqueness of weak solutions for the generalized stochastic Navier-Stokes-Voigt equations |
| topic | Probability |
| url | https://arxiv.org/abs/2403.08001 |