Global Small-Time Approximate Null and Lagrangian Controllability of the Viscous Non-Resistive MHD System in a 3 D Domain with Navier Type Boundary Conditions

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 [7] to establish the global small-time null controllability of the incompressible Navier-Stokes equations in the class of weak solutions and extended in [34] 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. Although 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 \(H^8\).