The dissertation consists of two projects on the regularity of the three-dimensional incompressible Navier-Stokes equations. In the first project, we study Navier-Stokes regularity on the half-space. The existence of minimal blowup-generating initial data, under the assumption that there exists an initial data leading to finite-time singularity, has been studied by Rusin and Sverak (2011), Jia and Sverak (2013), and Gallagher, Koch and Planchon (2013, 2016) in several critical spaces on the whole space. Our aim is to study the influence of the boundary on the existence of minimal blowup data. We introduce a type of weighted critical spaces for the external force that is better-suited for our analysis than the usual Lebesgue spaces. We reestablish regularity theory for the Stokes equations and local-in-time regularity for the Navier-Stokes equations. Our main tools to treat regularity near the boundary are the notion of "split'' weak solutions introduced by Seregin and Sverak (2017), the boundary regularity criteria and special decomposition of the pressure near the boundary due to Seregin (2002). Our method works well for both the half-space and the whole space. Our second project is motivated by the work of Li (2014). He introduces a hypothetical relation between the mesh size and the size of the corresponding numerical solution which guarantees the global existence of the exact solution. We formulate this problem for a continuous setting and identify some key difficulties.