On the Strong Solvability of the Navier-Stokes Equations

In this paper we study the strong solvability of the Navier-Stokes equations for rough initial data. We prove that there exists essentially only one maximal strong solution and that various concepts of generalized solutions coincide. We also apply our results to Leray-Hopf weak solutions to get improvements over some known uniqueness and smoothness theorems. We deal with rather general domains including, in particular, those having compact boundaries.