LOCAL WELL-POSEDNESS OF FREE SURFACE PROBLEMS FOR THE NAVIER-STOKES EQUATIONS IN A GENERAL DOMAIN

In this paper, we prove the local well-posedness of the free boundary problems of Navier-Stokes equations in a general domain Omega subset of R-N (N >= 2). The velocity field is obtained in the maximal regularity class W-q,p(2,1)(Omega x (0,T)) = L-p((0,T),W-q(2) (Omega) (N)) boolean AND W-p(1)((0,T); L-q (Omega) (N)) (2 < p < infinity and N < q < infinity) for any initial data satisfying certain compatibility conditions. The assumption of the domain Omega is the unique existence of solutions to the weak Dirichlet-Neumann problem as well as some uniformity of covering of the closure of Omega. A bounded domain, a perturbed half space, and a perturbed layer satisfy the conditions for the domain, and therefore drop problems and ocean problems are treated in the uniform manner. Our method is based on the maximal L-p-L-q regularity theorem of a linearized problem in a general domain.