Mathematical analysis of a levitation model

Abstract

The aim of this paper is to study a model of electromagnetic levitation for a metallic rigid body. The model is constituted by the transient linear model of eddy currents under the hypothesis of axisymmetry, written in terms of a magnetic potential vector, coupled with an ODE which governs the vertical motion of the body. The electromagnetic model is a parabolic-elliptic PDE which parabolicity region is the position occupied by the body, which changes with time. Besides, Lorentz force appears in the RHS of the ODE. Thus, the model exhibits a coupling of geometrical nature. We establish the existence and uniqueness of solution of the coupled problem and we study its maximally defined solution. In particular, we prove that a blow-up of the velocity of the body cannot happen. Our techniques involve: a reformulation of the coupled problem as a causal differential equation, an adaptation of the theory about this kind of equations and a result of locally Lipschitz dependence of the magnetic potential vector with respect to the velocity of the body.