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. Our main tools to treat regularity near the boundary are (1) the notion of ``split'' weak solutions introduced by Seregin and Sverak (2017), (2) the boundary epsilon-regularity criteria and (3) a 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. Joint work with Vladimir Sverak.