The Stokes and Navier-Stokes equations in layer domains with and without a free surface

This thesis is concerned with certain aspects of the Stokes- and Navier-Stokes equations in layer domains with and without a free surface. We investigate the Stokes equations in layer domains in the endpoints L1 and Linfty of the scale of Lebesgue spaces Lp and show that the Stokes operator in solenoidal subspaces of L1 and Linfty generates a holomorphic semigroup if and only if the spacial dimension of the layer dimension is two. In the last chapter we investigate the singular limit of vanishing surface tension for a free boundary problem for the Navier-Stokes equations and show convergence of solutions in the Lp maximal regularity space.