Mixed velocity, stress, current, and potential boundary conditions for stationary MHD flow

This paper is concerned with a boundary-value problem, describing the stationaryflow of a viscous, incompressible, electrically conducting fluid, confined to a bounded region of space, under mixed boundary conditions. The flow is governed by the Navier-Stokes equations, Ohmʹs law, and the Biot-Savart law; the boundary conditions involve the velocity field, stress tensor, electric current density, and electric potential. We derive a mixed variational formulation of the problem, which lends itself naturally to finite-element discretizations, and prove the existence and uniqueness of a (small) solution under the assumption of sufficiently small data.