## On local solutions to a free boundary problem for incompressible viscous magnetohydrodynamics in the $L_p$-approach

### Volume 566 / 2021

#### Abstract

We consider the motion of incompressible magnetohydrodynamics (mhd) 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 electromagnetic fields are imposed. We can distinguish two cases which have an essential influence on the proofs of existence: no jump of the magnetic field (Part 2) and a jump (Part 1). In the no jump case we prove local existence of solutions such that the velocity and the magnetic field belong to $W_r^{2,1}$, $r \gt 3$. In the case of any jump of the magnetic field we prove existence of local solutions such that the velocity belongs to $W_r^{3,3/2}$ and the magnetic field to $W_r^{2,1}$, $r \gt 5/2$.