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On the controllability of the Navier-Stokes equation in a rectangle, with a little help of a distributed phantom force

Jean-Michel Coron, Frédéric Marbach, Franck Sueur, Ping Zhang

TL;DR

We study controllability of the 2D Navier–Stokes equation in a rectangle $\Omega=(0,L)\times(-1,1)$ with controls acting on the vertical boundaries and no prescribed data on the horizontal walls. The approach combines a vorticity-flushing strategy, a boundary-layer/Prandtl-type correction, an analytic-in-tangential-variables framework (Cauchy–Kowalevski style) to manage derivative loss, and a Chemin-type scheme to handle nonlinearity, all organized around a carefully scaled, asymptotic construction. The main result shows that for any initial data $u_*=\in L^2_{\mathrm{div}}(\Omega)$, any $k\in\mathbb{N}$ and any $\eta>0$, there exists a forcing $f\in L^1((0,T);H^k(\Omega))$ with $\|f\|_{L^1((0,T);H^k(\Omega))}\le\eta$ and a weak Leray solution with $u(0)=u_*$ and $u(T)=0$, achieving near-null controllability with arbitrarily small forcing. However, extending this to the true $f=0$ case remains challenging due to boundary-layer amplification, motivating the use of phantom forcing to regularize and localize actions; this yields a constructive, near-global controllability result in a flat rectangle and advances Lions’ program while highlighting the central role of boundary layers in such problems.

Abstract

This note echoes the talk given by the second author during the Journées EDP 2018 in Obernai. Its aim is to provide an overview and a sketch of proof of the result obtained by the authors, concerning the controllability of the Navier-Stokes equation. We refer the interested readers to the original paper for the full technical details of the proof, which will be omitted here, to focus on the main underlying ideas.

On the controllability of the Navier-Stokes equation in a rectangle, with a little help of a distributed phantom force

TL;DR

We study controllability of the 2D Navier–Stokes equation in a rectangle with controls acting on the vertical boundaries and no prescribed data on the horizontal walls. The approach combines a vorticity-flushing strategy, a boundary-layer/Prandtl-type correction, an analytic-in-tangential-variables framework (Cauchy–Kowalevski style) to manage derivative loss, and a Chemin-type scheme to handle nonlinearity, all organized around a carefully scaled, asymptotic construction. The main result shows that for any initial data , any and any , there exists a forcing with and a weak Leray solution with and , achieving near-null controllability with arbitrarily small forcing. However, extending this to the true case remains challenging due to boundary-layer amplification, motivating the use of phantom forcing to regularize and localize actions; this yields a constructive, near-global controllability result in a flat rectangle and advances Lions’ program while highlighting the central role of boundary layers in such problems.

Abstract

This note echoes the talk given by the second author during the Journées EDP 2018 in Obernai. Its aim is to provide an overview and a sketch of proof of the result obtained by the authors, concerning the controllability of the Navier-Stokes equation. We refer the interested readers to the original paper for the full technical details of the proof, which will be omitted here, to focus on the main underlying ideas.

Paper Structure

This paper contains 21 sections, 1 theorem, 27 equations, 2 figures.

Table of Contents

  1. Geometric setting
  2. Cauchy problem
  3. A conjecture of Lions
  4. Our main controllability result
  5. Discussion on earlier results
  6. Small initial data and local results
  7. Global results without boundaries
  8. Navier slip-with-friction boundary condition
  9. Partial results with large forcing
  10. A strategy based on the flushing of the vortexes
  11. Asymptotic implementation of the flushing method
  12. Tangential boundary layers
  13. Main difficulties
  14. Recasting amplification as a loss of derivative
  15. Cauchy-Kowaleskaya schema
  16. ...and 6 more sections

Key Result

Theorem 4.1

Let $T > 0$ and $u_* \in L^2_{\mathop{\mathrm{div}}\nolimits}(\Omega)$. For every $k \in \mathbb{N}$ and every $\eta > 0$, there exists a force $f \in L^1((0,T); H^k(\Omega))$ satisfying and an associated weak Leray solution $u \in C^0([0,T];L^2_{\mathop{\mathrm{div}}\nolimits}(\Omega)) \cap L^2((0,T);H^1(\Omega))$ to ns and dirichlet satisfying $u(0) = u_*$ and $u(T) = 0$.

Figures (2)

  • Figure 1: Physical domain $\Omega$
  • Figure 2: Flushing process for the vorticity

Theorems & Definitions (2)

  • Conjecture 3.1
  • Theorem 4.1