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Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditions

Manuel Rissel, Ya-Guang Wang

TL;DR

This work establishes small-time global approximate controllability for incompressible MHD in smooth bounded domains with boundary control on a portion of the boundary and linearly coupled Navier slip-with-friction conditions on the remainder. The authors integrate the return method with the well-prepared dissipation approach, using domain extensions and Elsasser variables to manage the coupled velocity and magnetic-field dynamics. A detailed boundary-layer analysis shows how dissipation in the boundary layers can be harnessed to drive the system toward prescribed smooth states, including a second result with a pressure-like term $q$ in the induction equation under broader boundary data. The results advance the controlled MHD theory in nontrivial geometries and boundary couplings, highlighting the importance of planar simply-connected domains for eliminating the pressure term and guiding future extensions to 3D settings and more general boundary interactions.

Abstract

We study the small-time global approximate controllability for incompressible magnetohydrodynamic (MHD) flows in smoothly bounded two- or three-dimensional domains. The controls act on arbitrary nonempty open portions of each connected boundary component, while linearly coupled Navier slip-with-friction conditions are imposed along the uncontrolled parts of the boundary. Some choices for the friction coefficients give rise to interacting velocity and magnetic field boundary layers. We obtain sufficient dissipation properties of these layers by a detailed analysis of the corresponding asymptotic expansions. For certain friction coefficients, or if the obtained controls are not compatible with the induction equation, an additional pressure-like term appears. We show that such a term does not exist for problems defined in planar simply-connected domains and various choices of Navier slip-with-friction boundary conditions.

Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditions

TL;DR

This work establishes small-time global approximate controllability for incompressible MHD in smooth bounded domains with boundary control on a portion of the boundary and linearly coupled Navier slip-with-friction conditions on the remainder. The authors integrate the return method with the well-prepared dissipation approach, using domain extensions and Elsasser variables to manage the coupled velocity and magnetic-field dynamics. A detailed boundary-layer analysis shows how dissipation in the boundary layers can be harnessed to drive the system toward prescribed smooth states, including a second result with a pressure-like term in the induction equation under broader boundary data. The results advance the controlled MHD theory in nontrivial geometries and boundary couplings, highlighting the importance of planar simply-connected domains for eliminating the pressure term and guiding future extensions to 3D settings and more general boundary interactions.

Abstract

We study the small-time global approximate controllability for incompressible magnetohydrodynamic (MHD) flows in smoothly bounded two- or three-dimensional domains. The controls act on arbitrary nonempty open portions of each connected boundary component, while linearly coupled Navier slip-with-friction conditions are imposed along the uncontrolled parts of the boundary. Some choices for the friction coefficients give rise to interacting velocity and magnetic field boundary layers. We obtain sufficient dissipation properties of these layers by a detailed analysis of the corresponding asymptotic expansions. For certain friction coefficients, or if the obtained controls are not compatible with the induction equation, an additional pressure-like term appears. We show that such a term does not exist for problems defined in planar simply-connected domains and various choices of Navier slip-with-friction boundary conditions.
Paper Structure (56 sections, 26 theorems, 337 equations, 7 figures)

This paper contains 56 sections, 26 theorems, 337 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Navier slip-with-friction boundary conditions.
  3. Main results
  4. The class $\mathbf{S}$.
  5. Related literature and organization of the article
  6. Organization of this article.
  7. Preliminaries
  8. Domain extensions
  9. Function spaces
  10. Initial data extensions
  11. Weak controlled trajectories
  12. The case of ${\rm H}(\mathcal{E})$ data (e.g., the first main theorem
  13. Definition of weak controlled trajectories.
  14. Existence of weak solutions.
  15. A pressure-like gradient in the induction equation
  16. ...and 41 more sections

Key Result

Theorem 1.2

Assume that $\Omega \subset \mathbb{R}^2$ is simply-connected, that $\bm{M}_2 = \bm{L}_2 = \bm{0}$, and that $\Gamma_{\operatorname{c}}\subset\Gamma$ is connected. Then, for arbitrarily fixed $T_{\operatorname{ctrl}} > 0$, $\delta > 0$, and $\bm{u}_0, \bm{B}_0, \bm{u}_1, \bm{B}_1 \in {\rm L}^{2}_{\o to the MHD equations equation:MHD00 which obeys the terminal condition

Figures (7)

  • Figure 1: Sketch of a smoothly bounded simply-connected domain $\Omega \subset \mathbb{R}^2$ with connected controlled boundary $\Gamma_{\operatorname{c}}$, which is indicated by a dashed line.
  • Figure 2: Two exemplary domains that are covered by \ref{['theorem:main']}.
  • Figure 3: A multiply-connected domain $\Omega \subset \mathbb{R}^2$ with two controlled boundary components and extension $\mathcal{E}$. The dashed lines mark the controlled boundaries.
  • Figure 4: Sketch of an annulus $\mathcal{E}$ with control region $\omega$ as required by \ref{['theorem:annulus']}. In order to integrate \ref{['theorem:annulus']} into the notational framework of Theorems \ref{['theorem:main1']} and \ref{['theorem:main']}, one can identify $\Omega = \mathcal{E}\setminus \omega$.
  • Figure 5: A sketch of a cylindrical domain. The arrows indicate the profile $\bm{z}^0$, which is constant in the original domain, hence curl-free and divergence-free, but ceases to be divergence-free at certain parts in the extended domain.
  • ...and 2 more figures

Theorems & Definitions (67)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Example 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 57 more