Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface

In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the L_p in time L_q in space maximal regularity class, (2<p<∞, N<q<∞, and 2/p+N/q<1), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the L_p-L_q framework for the linearized equations with the help of maximal L_p-L_q regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.