In view of the lack of a global regularity theorem for the solutions $(v,p)$ of the Navier-Stokes equations there has been a great deal of activity in establishing sufficient conditions on the velocity $v$ in order to guarantee the regularity of the solution. However, nontrivial conditions involving the pressure seem not to be available in the literature. In this paper we present a sharp sufficient condition involving a combination of $v$ and $p$. The proof relies on the truncation method, introduced in reference  for studying scalar elliptic equations and developed further by many authors (see, in particular  and ). In the sequel we use some basic results proved in .
"Concerning the regularity of the solutions to the Navier-Stokes equations via the truncation method. I." Differential Integral Equations 10 (6) 1149 - 1156, 1997. https://doi.org/10.57262/die/1367438225