Suppose u solves the incompressible Navier-Stokes equations on R^d and is bounded by a constant C in some critical space (e.g., L^d) uniformly in time. In this talk we discuss the problem of controlling the solution's regularity explicitly in terms of C, which can shed new light on qualitative regularity theorems such as that of Escauriaza-Seregin-Sverak and potential blow-up phenomena.

This work is partly joint with Wojciech Ozanski.