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Existence and uniqueness of weak solutions for the generalized stochastic Navier-Stokes-Voigt equations

Ankit Kumar, Hermenegildo Borges de Oliveira, Manil T. Mohan

Abstract

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.

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: \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 , accounts for the velocity field, is the pressure, is a body force and the final term stay for the stochastic forces. Here, and are given positive constants that account for the kinematic viscosity and relaxation time, and the power-law index 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.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.
Paper Structure (20 sections, 13 theorems, 243 equations)

This paper contains 20 sections, 13 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Literature review
  4. Objectives and novelties of the paper
  5. Organization
  6. Auxiliary results
  7. Function spaces
  8. Stochastic setting
  9. Probabilistic weak formulation
  10. Assumptions and main result
  11. Pressure decomposition
  12. Solvability of The Approximate System
  13. Approximate solutions
  14. Uniform estimates
  15. Weak convergence
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let the map $\mathbf{M}:[0,T]\times\mathbb{R}^d\times\Omega\longrightarrow\mathcal{M}^{d\times d}(\mathbb{R})$, defined by $(t, x, \omega)\mapsto \mathbf{M}=\mathbf{M}(t, x, \omega)$, and $\mathbf{b}:[0,T]\times\mathbb{R}^d\times\Omega\longrightarrow\mathbb{R}^d$, defined by $(t, x, \omega)\mapsto \ where $K_t(R)$ is an $\mathbb{R}^+-$valued process adapted to the filtration $\{\mathscr{F}_t\}_{t\

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4: Probabilistically strong solution
  • Definition 2.5: Pathwise uniqueness
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 3.1
  • ...and 31 more