A free boundary problem for an incompressible viscous fluid in a domain with noncornpact boundaries is considered; the upper boundary is to be determined by equilibrium conditions involving the fluid stress tensor and its surface tension. It is proved that if the data of the problem are regular, then the free boundary, the velocity vector and the pressure are regular. Furthermore the exponential decay of the solution is shown.
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R.S. Gellrich, Free Boundary Value Problems for the Stationary Navier-Stokes Equations in Domains with Noncompact Boundaries. Z. Anal. Anwend. 12 (1993), no. 3, pp. 425–455DOI 10.4171/ZAA/554