We consider the problem of evolution of a finite isolated mass of a viscous incompressible liquid with a free surface. We assume that the initial configuration of the liquid hasn arbitrary shape, the initial free boundary possesses a certain regularity and the initial velocity satisfies only natural compatibility and regularity conditions (but its smallness is not assumed). We prove that this problem is well posed, i.e., we construct a local in time solution belonging to some Sobolev–Slobodetskii spaces. We expect that this result can be helpful for the analysis of more complicated problems, for instance, problems of magnetohydrodynamics. Bibliography: 9 titles.
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Translated from Problems in Mathematical Analysis 50, September 2010, pp. 87–112
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Padula, M., Solonnikov, V.A. On the local solvability of free boundary problem for the Navier–Stokes equations. J Math Sci 170, 522–553 (2010). https://doi.org/10.1007/s10958-010-0099-3
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DOI: https://doi.org/10.1007/s10958-010-0099-3