Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stochastic-periodic Homogenization of Non-stationary Incompressible Navier-Stokes Type Equations

Tchinda Franck, Fotso Tachago Joel, Dongho Joseph

Abstract

In this paper, we study the stochastic-periodic homogenization of Non-stationary Navier-Stokes Type Equations on anisotropic heterogeneous media. More precisely, we are interested in the stochastic-periodic homogenization of its variational formulation. This problematic relies on the notion of dynamical system. It is shown by the stochastic two-scale convergence method that the resulting homogenized limit equation is of the same form of this variational formulation with suitable coefficients.

Stochastic-periodic Homogenization of Non-stationary Incompressible Navier-Stokes Type Equations

Abstract

In this paper, we study the stochastic-periodic homogenization of Non-stationary Navier-Stokes Type Equations on anisotropic heterogeneous media. More precisely, we are interested in the stochastic-periodic homogenization of its variational formulation. This problematic relies on the notion of dynamical system. It is shown by the stochastic two-scale convergence method that the resulting homogenized limit equation is of the same form of this variational formulation with suitable coefficients.
Paper Structure (8 sections, 10 theorems, 74 equations)

This paper contains 8 sections, 10 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Fundamentals of stochastic two-scale convergence
  3. Preliminary on dynamical systems and ergodic theory
  4. stochastic two-scale convergence
  5. Homogenized Navier-Stokes system
  6. The $\epsilon$-model for (\ref{['tc11']})
  7. Existence of solutions to the weak formulation of (\ref{['tc1']})-(\ref{['tc7']})
  8. Main homogenization results for (\ref{['tc9']})-(\ref{['tc10']})

Key Result

Lemma 2.1

Let $\mathbf{v} = (v_{i}) \in L^{p}(\Omega)^{N}$ such that $\sum_{i=1}^{N} \int_{\Omega} \mathbf{v}\cdot\mathbf{g}\,d\mu = 0$ for all $\mathbf{g} \in \mathcal{V}^{\omega}_{\textup{div}} = \{ \mathbf{f} = (f_{i})\in \mathcal{C}^{\infty}(\Omega)^{N} : \textup{div}_{\omega}\mathbf{f}=0 \}$. Then $\text

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Theorem 2.8
  • Proposition 2.9
  • ...and 6 more