Geodesic
Solvability of a~free boundary problem of magnetohydrodynamics in an infinite time interval
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 43, Tome 410 (2013), pp. 131-167

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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 $W_2^{2+l,1+l/2}$, $1/2$. 
@article{ZNSL_2013_410_a5,
     author = {V. A. Solonnikov and E. V. Frolova},
     title = {Solvability of a~free boundary problem of magnetohydrodynamics in an infinite time interval},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {131--167},
     publisher = {mathdoc},
     volume = {410},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_410_a5/}
}
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V. A. Solonnikov; E. V. Frolova. Solvability of a~free boundary problem of magnetohydrodynamics in an infinite time interval. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 43, Tome 410 (2013), pp. 131-167. http://geodesic.mathdoc.fr/item/ZNSL_2013_410_a5/