Existence of Local Solutions to a Free Boundary Problem for Incompressible Viscous Magnetohydrodynamics

We consider the motion of an incompressible magnetohydrodynamics with resistivity in a domain bounded by a free surface which is coupled through the free surface with an electromagnetic field generated by a magnetic field prescribed on an exterior fixed boundary. On the free surface, transmission conditions for the electromagnetic field are imposed. As transmission condition we assume jumps of tangent components of magnetic and electric fields on the free surface. We prove local existence of solutions such that velocity and magnetic fields belong to \(H^{2+\alpha ,1+\alpha /2}\) $H^{2+\alpha ,1+\alpha /2}$ , \(\alpha >5/8\) $\alpha >5/8$ .