We prove small-time existence of strong solutions of a free-boundary-value problem, which describes the motion of an incompressible viscous fluid occupying a semi-infinite domain bounded above by a free surface. This problem was studied by Beale  and others in $L^2$-Sobolev spaces. In contrast to the latter contribution we study solutions in $L^q$-Sobolev spaces for $q>n$ in space dimension $n\geq 2$. This approach has the advantage that the regularity assumptions can be reduced in comparison to . In order to solve the linearized system, we use the nonstationary reduced Stokes equations with a mixed boundary condition and the maximal regularity of the associated reduced Stokes operator.
"The initial-value problem for the Navier-Stokes equations with a free surface in $L^q$-Sobolev spaces." Adv. Differential Equations 10 (1) 45 - 64, 2005.