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Uniform local null control of the Leray-$α$ model

Fágner D. Araruna, Enrique Fernández-Cara, Diego A. Souza

Abstract

This paper deals with the distributed and boundary controllability of the so called Leray-$α$ model. This is a regularized variant of the Navier-Stokes system ($α$ is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-$α$ equations are locally null controllable, with controls bounded independently of $α$. We also prove that, if the initial data are sufficiently small, the controls converge as $α\to 0^+$ to a null control of the Navier-Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.

Uniform local null control of the Leray-$α$ model

Abstract

This paper deals with the distributed and boundary controllability of the so called Leray- model. This is a regularized variant of the Navier-Stokes system ( is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray- equations are locally null controllable, with controls bounded independently of . We also prove that, if the initial data are sufficiently small, the controls converge as to a null control of the Navier-Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.
Paper Structure (15 sections, 16 theorems, 111 equations, 1 figure)

This paper contains 15 sections, 16 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction. The main results
  2. Preliminaries
  3. The Stokes operator
  4. Well-posedness for the Leray-$\alpha$ system
  5. Carleman inequalities and null controllability
  6. The distributed case: Theorems \ref{['NC-LERAY']} and \ref{['CONVERGENCE']}
  7. The boundary case: Theorems \ref{['NC-LERAY-BOUNDARY']} and \ref{['CONVERGENCE-BOUNDARY']}
  8. Additional comments and questions
  9. Controllability problems for semi-Galerkin approximations
  10. Another strategy: applying an inverse function theorem
  11. On global controllability properties
  12. The Burgers-$\alpha$ system
  13. Local exact controllability to the trajectories
  14. Controlling with few scalar controls
  15. Other related controllability problems

Key Result

Theorem 1

There exists $\epsilon>0~(\hbox{independent of}~\alpha)$ such that, for each $\mathbf{y}_0 \in \mathbf{H}$ with $\|\mathbf{y}_0\| \leq \epsilon$, there exist controls $\mathbf{v}_\alpha\in L^\infty(0,T;\mathbf{L}^2(\omega))$ such that the associated solutions to CLalpha fulfill null_condition. Fur where $C$ is also independent of $\alpha$.

Figures (1)

  • Figure 1: The domain $\widetilde{\Omega}$

Theorems & Definitions (18)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • ...and 8 more