Global existence of solutions of the free boundary problem for the equations of magnetohydrodynamic imcompressible viscous fluid (Q1772556)

The title problem is investigated for a magnetic fluid occupying a domain \(\Omega_t\subset\mathbb{R}^3\) bounded by a free surface \(S_t\). In a domain \(D_t\subset\mathbb{R}^3\) which is exterior to \(\Omega_t\), there is a gas under constant pressure. Moreover, in the domain \(D_t\) there exists an electromagnetic field generated by some currents which are located on a fixed exterior boundary \(B\) of \(D_t\). The original problem is reformulated in Lagrangian coordinates, and, using some estimates obtained via Korn inequality, the author proves the existence of global solution for sufficiently small initial velocity and magnetic field.