| This thesis has two parts. In the first part, we establish the equivalence of two Sobolev spaces of divergence-free vector fields in open bounded domains in R2 . We then generalize the results to open, bounded, and axisymmetric domains in R3 . The key to these results is a theorem on Sobolev spaces by Hedberg, the technique of stream functions, and the topological structure of R2 .;In the second part, we apply the results of the first part to Optimal Shape Design problems. We consider submarines in two-dimensional NSE flows with a given volume and prove the existence of an drag-minimizing shape in a suitable set of shapes. |
