The initial-value problem for the Navier-Stokes equations with a free surface in $L^q$-Sobolev spaces

Abstract

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 [6] 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 [6]. 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.