Strong Solutions to the Stokes Problem with Navier Slip on a Wedge

Abstract

In this thesis we consider the incompressible and stationary Stokes problem with Navier-slip boundary conditions on an infinite two-dimensional wedge with opening angle θ. As is common for differential equations on domains with corners, the problem is decomposed into a singular expansion near the corner (polynomial problem) and a regular remainder (smooth problem). We prove existence and uniqueness of solutions to the smooth problem related to the Stokes equation which is given by -PΔu = f, where P is the Helmholtz projection. By means of the Lax-Milgram theorem it is found that this problem has a unique strong solution in a certain class of weighted Sobolev spaces if the opening angle θ is small enough. Direct application of the Lax-Milgram theorem would normally only yield a weak solution. However, by introducing additional bilinear forms we gain control on all second order derivatives and therewith obtain a strong solution. Finally, we touch upon the time-dependent Stokes problem and the polynomial problem.