Existence and Asymptotic Behavior of Solutions to the Time-Periodic Navier-Stokes Equations in a Layer Domain with Nonhomogeneous Boundary Data

This dissertation is dedicated to the analysis of the Navier-Stokes equations in a timeperiodic framework in the so-called layer domain Π = R2 × (0, 1), described by: ∂tu − νΔu + (u · ∇)u + ∇p = f in [0, T] × Π, div u = 0 in [0, T] × Π, u|∂Π = a for all t ∈ [0, T] , u|t=0 = u|t=T in Π. The velocity field u and the pressure p are unknowns, while the external force f is prescribed. Challenges arise due to unboundedness of the layer Π and from introduction of a nonhomogeneous boundary condition a. The investigated topics regarding this system of differential equations are the theory of existence and the theory of asymptotics. In the existence theory a case distinction with respect to the boundary condition has to be made: For boundary values having zero flux – where flux is the balance of in- and out-flow through the boundary – existence of solutions is proved without restrictions on the (size of the) data. In the case of non-zero flux a statement of existence is achieved for boundary values being small in a certain norm. The theory of asymptotics is concerned with the behavior of solutions towards spatial infinity. At first, the linear Stokes system is analyzed, continuing the work of Pileckas and Specovius-Neugebauer in [42]. An asymptotic representation for solutions to this problem is derived, which is a generalization of Pileckas and Specovius-Neugebauer’s main result. Then, in investigations of the non-linear Navier-Stokes equations, this theorem is employed to prove an asymptotic representation for solutions to the nonlinear system as well, where the leading term in fact coincides with that of the Stokes problem.