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Global small-time approximate null and Lagrangian controllability of the viscous non-resistive MHD system in a $3D$ domain with Navier type boundary conditions

Jiajiang Liao, Franck Sueur, Ping Zhang

TL;DR

This work establishes global small-time approximate controllability for the viscous non-resistive MHD system in a 3D bounded domain with Navier-type boundaries, achieving approximate null states in $H^1$ and $L^ fty$ for both velocity and magnetic field. The authors extend the Coron return method to the MHD setting and develop a multi-scale, boundary-layer framework that couples fast-time controls with higher-order expansions, all within conormal Sobolev spaces to handle boundary effects. A key contribution is the construction of a detailed approximate solution and rigorous remainder estimates showing the residual remains $o(1)$ as the fast scale parameter $ o 0$, thereby achieving global small-time approximate controllability. They also extend the Lagrangian controllability concept to MHD, proving that a flow map can transport a Jordan surface to a nearby target within prescribed tolerance, with improved regularity under smoother initial data. These results advance boundary-control theory for coupled hyperbolic–parabolic systems without magnetic diffusion and have potential implications for manipulating conducting fluids in bounded domains.

Abstract

We consider the incompressible viscous MHD system without magnetic diffusion in a $3D$ bounded domain with Navier type boundary condition. We establish the global small-time approximate null controllability and the Lagrangian controllability of the system, in the class of smooth solutions, by following the approach initiated in \cite{CMS} to establish the global small-time null controllability of the incompressible Navier-Stokes equations in the class of weak solutions and extended in \cite{LSZ1} to establish the global small-time null and Lagrangian controllability of the incompressible Navier-Stokes equations in the class of strong solutions. This approach makes use of controls with an extra fast scale in time and some corresponding multi-scale asymptotic expansions of the controlled solution. This expansion is constructed by an iterative process which requires some regularity. The extra-difficulty here is that the MHD system at stake is mixed hyperbolic-parabolic, without any regularizing effect on the magnetic field. Despite our strategy makes use of a quite precise asymptotic expansion, we succeed to cover the case where the initial velocity belongs the Sobolev space $H^{24}$ and the initial magnetic field belongs to the Sobolev space $H^8$.

Global small-time approximate null and Lagrangian controllability of the viscous non-resistive MHD system in a $3D$ domain with Navier type boundary conditions

TL;DR

This work establishes global small-time approximate controllability for the viscous non-resistive MHD system in a 3D bounded domain with Navier-type boundaries, achieving approximate null states in and for both velocity and magnetic field. The authors extend the Coron return method to the MHD setting and develop a multi-scale, boundary-layer framework that couples fast-time controls with higher-order expansions, all within conormal Sobolev spaces to handle boundary effects. A key contribution is the construction of a detailed approximate solution and rigorous remainder estimates showing the residual remains as the fast scale parameter , thereby achieving global small-time approximate controllability. They also extend the Lagrangian controllability concept to MHD, proving that a flow map can transport a Jordan surface to a nearby target within prescribed tolerance, with improved regularity under smoother initial data. These results advance boundary-control theory for coupled hyperbolic–parabolic systems without magnetic diffusion and have potential implications for manipulating conducting fluids in bounded domains.

Abstract

We consider the incompressible viscous MHD system without magnetic diffusion in a bounded domain with Navier type boundary condition. We establish the global small-time approximate null controllability and the Lagrangian controllability of the system, in the class of smooth solutions, by following the approach initiated in \cite{CMS} to establish the global small-time null controllability of the incompressible Navier-Stokes equations in the class of weak solutions and extended in \cite{LSZ1} to establish the global small-time null and Lagrangian controllability of the incompressible Navier-Stokes equations in the class of strong solutions. This approach makes use of controls with an extra fast scale in time and some corresponding multi-scale asymptotic expansions of the controlled solution. This expansion is constructed by an iterative process which requires some regularity. The extra-difficulty here is that the MHD system at stake is mixed hyperbolic-parabolic, without any regularizing effect on the magnetic field. Despite our strategy makes use of a quite precise asymptotic expansion, we succeed to cover the case where the initial velocity belongs the Sobolev space and the initial magnetic field belongs to the Sobolev space .
Paper Structure (32 sections, 29 theorems, 324 equations, 2 figures)

This paper contains 32 sections, 29 theorems, 324 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Main results
  4. Assumptions on the initial data
  5. Small-time global approximate null controllability
  6. Small-time global approximate Lagrangian controllability
  7. Related literature
  8. On the well-posedness of the non-resistive MHD system.
  9. On the Navier-Stokes equations
  10. On the MHD equations
  11. Status of the main results here VS the literature
  12. Scheme of the proof of Theorem \ref{['mainth']}
  13. Time scaling
  14. Return method
  15. Boundary layers and multi-scale asymptotic expansion
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $T>0$ and we assume that the initial data $(u_0,B_0)$ satisfies ini1 and ini2. Then for any $\epsilon>0$, there exists a solution $(u,p,B)$ of MHD with which satisfies

Figures (2)

  • Figure 1: Extension of the physical domain $\Omega\subset\mathcal{O}$
  • Figure 2: Domains for the Runge-type theorem

Theorems & Definitions (54)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6: Proposition 3.3 of LSZ1
  • Lemma 3.1: Lemma 4.1 of LSZ1
  • ...and 44 more