On solutions to Navier-Stokes equations in critical spaces.

Abstract

In this thesis, we consider solutions to the incompressible Navier-Stokes equa- tions in three spatial dimensions in the critical homogenous Sobolev space. We attempt to unify the theory of mild solutions and the theory of suitable weak solutions to show that assuming the existence of initial data leading to a finite time singularity the set of such initial is closed in the weak topology and sequen- tially compact modulo translations and dilations. This result is motivated by a theorem of Gallagher, Iftimie and Planchon which states that this set is closed in the strong topology. We present two proofs of our result. The first one is based on the theory of suitable weak solutions and partial regularity for the Navier-Stokes equaitons. The second approach is rooted in the profile decomposition developed by Gallagher.