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Lagrangian controllability in perforated domains

Mitsuo Higaki, Jiajiang Liao, Franck Sueur

TL;DR

This work addresses Lagrangian controllability for viscous incompressible flows in domains perforated by small holes under no-slip boundaries. It combines homogenization techniques (to Euler outside perforations or to Darcy in fully perforated regimes) with strong, localized forcing to achieve approximate transport of a polluted fluid patch to a target region, while keeping most particles outside the control zone. Central contributions include precise rate-dependent convergence results: 𝔭_E = min{α+β−3, α−3/2, β} governs the Euler limit in partially perforated domains, and 𝔭_D governs the Darcy limit in fully perforated domains, with explicit correctors w^ε, q^ε and a resistance matrix that quantify boundary-layer and perforation effects. The results demonstrate that, despite no-slip boundaries and complex microstructures, Lagrangian controllability can be achieved via homogenization and carefully designed controls, providing a rigorous pathway for steering flows in porous-like media with potential applications in engineering and geophysics.

Abstract

The question at stake in Lagrangian controllability is whether one can move a patch of fluid particles to a target location by means of remote action in a given time interval. In the last two decades, positive results have been obtained both for the incompressible Euler and Navier-Stokes equations. However, for the latter, the case where the fluid is contained within domains bounded by solid boundaries with the no-slip condition has not been addressed, with respect to the difficulty caused by viscous boundary layers. In this paper, we investigate the Lagrangian controllability of viscous incompressible fluid in perforated domains for which the fraction of volume occupied by the holes is sufficiently small. Moreover, we quantitatively distinguish situations depending on the parameters for holes (diameter and distance) and for fluid (size of the initial data). Our approach relies on recent results on homogenization for evolutionary problems and on weak-strong stability estimates in measure of flows, alongside classical results on Runge-type approximations for elliptic equations and on Cauchy-Kowalevsky-type theorems for equations with analytic coefficients. Here, homogenization refers to the vanishing viscosity limit outside a porous medium, where (after scaling in time) the Navier-Stokes equations are homogenized to the Euler or Darcy equations. Indeed, in the proof, we act on the Navier-Stokes equations by strong and fast forcing to leverage inviscid approximations, which is a standard technique in the theory of controllability.

Lagrangian controllability in perforated domains

TL;DR

This work addresses Lagrangian controllability for viscous incompressible flows in domains perforated by small holes under no-slip boundaries. It combines homogenization techniques (to Euler outside perforations or to Darcy in fully perforated regimes) with strong, localized forcing to achieve approximate transport of a polluted fluid patch to a target region, while keeping most particles outside the control zone. Central contributions include precise rate-dependent convergence results: 𝔭_E = min{α+β−3, α−3/2, β} governs the Euler limit in partially perforated domains, and 𝔭_D governs the Darcy limit in fully perforated domains, with explicit correctors w^ε, q^ε and a resistance matrix that quantify boundary-layer and perforation effects. The results demonstrate that, despite no-slip boundaries and complex microstructures, Lagrangian controllability can be achieved via homogenization and carefully designed controls, providing a rigorous pathway for steering flows in porous-like media with potential applications in engineering and geophysics.

Abstract

The question at stake in Lagrangian controllability is whether one can move a patch of fluid particles to a target location by means of remote action in a given time interval. In the last two decades, positive results have been obtained both for the incompressible Euler and Navier-Stokes equations. However, for the latter, the case where the fluid is contained within domains bounded by solid boundaries with the no-slip condition has not been addressed, with respect to the difficulty caused by viscous boundary layers. In this paper, we investigate the Lagrangian controllability of viscous incompressible fluid in perforated domains for which the fraction of volume occupied by the holes is sufficiently small. Moreover, we quantitatively distinguish situations depending on the parameters for holes (diameter and distance) and for fluid (size of the initial data). Our approach relies on recent results on homogenization for evolutionary problems and on weak-strong stability estimates in measure of flows, alongside classical results on Runge-type approximations for elliptic equations and on Cauchy-Kowalevsky-type theorems for equations with analytic coefficients. Here, homogenization refers to the vanishing viscosity limit outside a porous medium, where (after scaling in time) the Navier-Stokes equations are homogenized to the Euler or Darcy equations. Indeed, in the proof, we act on the Navier-Stokes equations by strong and fast forcing to leverage inviscid approximations, which is a standard technique in the theory of controllability.

Paper Structure

This paper contains 15 sections, 19 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lagrangian controllability in partially perforated domains
  3. Notation
  4. Controllability of Euler equations in torus
  5. Homogenization to Euler equations
  6. Correctors in partially perforated domains
  7. Rate of convergence
  8. Proof of Theorem \ref{['thm.main.E']}
  9. Lagrangian controllability in fully perforated domains
  10. Notation
  11. Controllability of Darcy equations in torus
  12. Homogenization to Darcy equations
  13. Proof of Theorem \ref{['thm.main.D']}
  14. Definitions from differential geometry
  15. Flows on a perforated domain

Key Result

Theorem 2.1

Let $\alpha>3/2$ and choose $\beta>0$ so that $3-\alpha < \beta < \alpha$, which ensures that $\mathfrak p_E$ in def.pE is positive. For given $0<\eta<1$, we define where $0<L<1$ refers to the length of $K$, $\lfloor\,\cdot\,\rfloor$ the floor function. Then, for any $0 < \varepsilon \le \varepsilon_0$ of the form def.eps and initial data $U^\varepsilon_0\in L^2_\sigma(\Omega^\varepsilon)$ satisf

Figures (2)

  • Figure 1: on the left: $K$, $\Omega_c$ and $P_1$; on the right: zoom on $K$, with $P_0^\varepsilon$ and $(\mathcal{T}^\varepsilon_i)_i$
  • Figure 2: position at time $\varepsilon^\beta$ of the particles initially in $P_0^\varepsilon$

Theorems & Definitions (23)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 13 more