A Generalization of Some Regularity Criteria to the Navier–Stokes Equations Involving One Velocity Component

We present generalizations of results concerning conditional global regularity of weak Leray–Hopf solutions to incompressible Navier–Stokes equations presented by Zhou and Pokorný in articles (Pokorný, Electron J Differ Equ (11):1–8, 2003; Zhou, Methods Appl Anal 9(4):563–578, 2002; Zhou, J Math Pure Appl 84(11):1496–1514, 2005); see also Neustupa et al. (Quaderni di Matematica, vol. 10. Topics in Mathematical Fluid Mechanics, 2002, pp. 163–183) We are able to replace the condition on one velocity component (or its gradient) by a corresponding condition imposed on a projection of the velocity (or its gradient) onto a more general vector field. Comparing to our other recent results from Axmann and Pokorný (A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component, in preparation), the conditions imposed on the projection are more restrictive here, however due to the technique used in Axmann and Pokorný (A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component, in preparation), there appeared a specific additional restriction on geometrical properties of the reference field, which could be omitted here.