Concerning the regularity of the solutions to the Navier-Stokes equations via the truncation method. I

Abstract

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 [3] for studying scalar elliptic equations and developed further by many authors (see, in particular [6] and [4]). In the sequel we use some basic results proved in [4].