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 paper proves that the 3D viscous non-resistive MHD system can be approximately controlled to zero in small time and that the flow can be controlled along trajectories, using multi-scale asymptotic expansions and controls with fast time scales.

Contribution

It extends controllability results from Navier-Stokes to the MHD system, handling the hyperbolic-parabolic mix without magnetic diffusion, with high regularity assumptions.

Findings

Global small-time approximate null controllability established.

Lagrangian controllability of the MHD system demonstrated.

Control strategies based on multi-scale asymptotic expansions successfully applied.

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 $H^8$.