On the non-stationary Navier-Stokes flows and reiterated homogenization

Technical Summary: On the Non-Stationary Navier-Stokes Type Flows and Reiterated Homogenization

Problem Statement
This paper investigates the deterministic reiterated homogenization of non-stationary (time-dependent) Navier-Stokes type equations defined on a fixed bounded domain $\Omega \subset \mathbb{R}^N$ (specifically $N=2$) over a time interval $]0, T[$. The governing equations describe fluid flow with periodically rapidly varying viscosity coefficients. The specific scaling of these coefficients involves two distinct microscopic scales, $\varepsilon$ and $\varepsilon^2$, leading to the definition:
$$a^\varepsilon_{ij}(x) = a_{ij}\left(\frac{x}{\varepsilon}, \frac{x}{\varepsilon^2}\right)$$
where $a_{ij}$ are continuous, bounded, symmetric, and uniformly elliptic functions defined on the product of two periodicity cells, $Y \times Z$. The study focuses on the Cauchy-Dirichlet boundary value problem for the velocity $u^\varepsilon$ and pressure $p^\varepsilon$, characterized by the equation:
$$\frac{\partial u^\varepsilon}{\partial t} + P^\varepsilon u^\varepsilon + \sum_{j=1}^N u^\varepsilon_j \frac{\partial u^\varepsilon}{\partial x_j} + \nabla p^\varepsilon = f, \quad \text{div } u^\varepsilon = 0$$
with zero initial and boundary conditions. The primary objective is to determine the macroscopic behavior of the solution $(u^\varepsilon, p^\varepsilon)$ as the parameter $\varepsilon \to 0$.

Methodology
The analysis relies on the method of reiterated two-scale convergence, an extension of the standard two-scale convergence technique designed to handle multiple scales of oscillation. The methodology proceeds through the following steps:

1. Functional Framework: The problem is formulated in appropriate Sobolev spaces, specifically $W(0, T)$, which embeds continuously into $C(0, T; H)$ and compactly into $L^2(0, T; H)$. The space $V$ (divergence-free vector fields) and its dual are utilized to handle the incompressibility constraint.
2. A Priori Estimates: Using the coercivity of the bilinear form associated with the operator $P^\varepsilon$ and the properties of the trilinear form representing the convective term, the authors establish uniform bounds for the sequence of solutions $(u^\varepsilon)$ in $W(0, T)$ and $(p^\varepsilon)$ in $L^2(0, T; L^2(\Omega))$.
3. Compactness and Convergence: Leveraging the compactness results of reiterated two-scale convergence (Theorems 1 and 2), the authors extract a subsequence converging to a limit triplet $(u_0, u_1, u_2)$. Here, $u_0$ represents the macroscopic limit, while $u_1$ and $u_2$ represent the first and second-order microscopic correctors associated with the scales $y = x/\varepsilon$ and $z = x/\varepsilon^2$, respectively.
4. Derivation of the Homogenized Model: By testing the variational formulation with specific oscillating test functions and passing to the limit, the authors derive a coupled system of equations. This involves solving local cell problems on the periodicity cells $Y$ and $Z$ to define corrector functions $\chi_{ik}$ and $\eta_{ik}$.
5. Corrector Analysis: Under additional regularity assumptions on the domain and data, the paper derives a corrector result, proving that the gradient of the original solution can be approximated by the sum of the gradients of the macroscopic solution and the scaled microscopic correctors.

Key Contributions and Results

• Macroscopic Homogenized Model: The paper derives a unique macroscopic limit problem (Theorem 5) of the Navier-Stokes type. The limit equation for the macroscopic velocity $u_0$ and pressure $p_0$ is:
  $$\frac{\partial u_0}{\partial t} + Q u_0 + \sum_{j=1}^N u^0_j \frac{\partial u_0}{\partial x_j} + \nabla p_0 = f$$
  where $Q$ is a homogenized differential operator with constant coefficients $q_{ijkh}$. These coefficients are explicitly defined via integrals involving the original coefficients $a_{ij}$ and the solutions to the cell problems on $Y$ and $Z$.
• Uniqueness: The paper establishes the uniqueness of the solution to the derived homogenized problem (Proposition 8), ensuring the well-posedness of the macroscopic model.
• Convergence Theorem: It is proven that $u^\varepsilon$ converges weakly to $u_0$ in $W(0, T)$ and that the gradients converge reiteratively in the two-scale sense to the sum of the macroscopic and microscopic gradients (Theorem 4).
• Corrector Result: A strong convergence result for the gradients is established (Theorem 6). Specifically, under suitable regularity assumptions, the error between the actual gradient $\nabla u^\varepsilon$ and the approximation $\nabla u_0 + \varepsilon \nabla_y u_1 + \varepsilon^2 \nabla_z u_2$ vanishes in the $L^2$ norm as $\varepsilon \to 0$.

Significance and Scope
The author positions this work as a contribution to the mathematical analysis of Navier-Stokes type flows with complex, multi-scale viscosity structures. The paper claims to be the first to address the reiterated homogenization of unsteady (non-stationary) nonlinear fluid flows in a deterministic setting with the specific scaling $a_{ij}(x/\varepsilon, x/\varepsilon^2)$.

The study is motivated by the need to model multiphase flows with spatially varying viscosities. While the author notes that the approach can be adapted to more general (non-periodic) behaviors of viscosities, the current work is strictly confined to the periodic setting.

The paper maintains a modest scope regarding time-dependent coefficients. The author explicitly states that the study assumes the viscosity coefficients $a_{ij}$ depend only on spatial variables, not time. The conclusion acknowledges that extending these results to time-dependent viscosities is a difficult task due to the lack of established uniqueness and estimates for such cases in the literature, noting that the current homogenization process remains valid provided those estimates and uniqueness results can be established.

The work builds upon the author's previous studies on steady-state flows and single-scale homogenization, as well as the foundational reiterated two-scale convergence theory developed by other researchers (cited as [5], [6], [8]). The results provide a rigorous mathematical foundation for approximating complex, rapidly oscillating fluid dynamics with a simpler, effective macroscopic model.