Inhomogeneous Boundary Value Problems for the Three-Dimensional Evolutionary Navier—Stokes Equations

In this paper, we study the solvability of inhomogeneous boundary value problems for the three-dimensional Oseen and Navier—Stokes equations in the following formulation: given function spaces for Dirichlet boundary conditions, initial values, and right-hand side forcing functions, find function spaces for solutions such that the operator generated by the boundary value problem for the Oseen equations establishes an isomorphism between the space of solutions and the spaces of data. Existence and uniqueness results for the solution of the time-dependent, three-dimensional Navier—Stokes equations are also established. These investigations are based on a theory of extensions of time-dependent, solenoidal vector fields that is developed in this paper. The results of this paper are indispensable to the study of optimal control problems with boundary control for the three-dimensional Navier—Stokes equations.