Solvability of a free boundary problem of magnetohydrodynamics in an infinite time interval

We prove global in time solvability of a free boundary problem governing the motion of a finite isolated mass of a viscous incompressible electrically conducting capillary liquid in vacuum, under the smallness assumptions on initial data. We assume that initial position of a free boundary is close to a sphere. We show that if $t\to\infty$, then the solution tends to zero exponentially and the free boundary tends to a sphere of the same radius, but, in general, the sphere may have a different center. The solution is obtained in Sobolev–Slobodetskii spaces $W_2^{2+l,1+l/2}$, $1/2<l<1$.