On the eigenvalues and eigenfunctions for a free boundary problem for incompressible viscous magnetohydrodynamics

The motion of incompressible magnetohydrodynamics (mhd) in a domain bounded by a free surface and coupled through it with an external electromagnetic field is considered. Transmission conditions for electric currents and magnetic fields are prescribed on the free surface. In this paper we show the idea of the proof of local existence by the method of successive approximations. For this we need linearized problems: the Stokes system for the velocity and pressure and the linear transmission problem for the electromagnetic field. We do not prove the local existence of solutions to the original problem but we show existence of a fundamental basis of functions for the linearized problems. Once we have such a basis, the existence of solutions to the linear problems can be shown by the Faedo–Galerkin method, as in other papers of Kacprzyk. The existence of solutions of the linear systems can also be shown by the method of regularizer.