Mixed Initial-Boundary Value Problem for the Three-Dimensional Navier-Stokes Equations in Polyhedral Domains

We study a mixed initial-boundary value problem for the Navier-Stokes equations, where the Dirichlet, Neumann and slip boundary conditions are prescribed on the faces of a three-dimensional polyhedral domain. We prove the existence, uniqueness and smoothness of the solution on a time interval $(0,T^{\ast })$, where $0<T^{\ast }\le T$.