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 RN (N 2). The velocity field is obtained in the maximal regularity class W2;1 q;p (0 T)) = Lp((0; T);Wq ()N) W¹ p ((0; T);Lq()N) (2< 1 and N < q < 1) for any initial data satisfying certain compatibility conditions. The assumption of the domain is the unique existence of solutions to the weak Dirichlet-Neumann problem as well as some uniformity of covering of the closure of. 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 Lp-Lq regularity theorem of a linearized problem in a general domain.