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

TL;DR

This paper establishes small-time global approximate controllability for incompressible MHD flows in bounded domains with boundary controls and Navier slip conditions, analyzing boundary layer interactions and dissipation properties.

Contribution

It provides new controllability results for MHD with coupled boundary conditions, including detailed boundary layer analysis and conditions for the absence of pressure-like terms.

Findings

Controllability achieved with boundary controls on open boundary parts.

Boundary layer dissipation analyzed via asymptotic expansions.

Pressure-like terms may be absent in planar simply-connected domains.

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.