A Sufficient Condition on the Pressure for the Regularity of Weak Solutions to the Navier-Stokes Equations

It is well known that a weak solution (v, p) to the Navier-Stokes equations is regular if v satisfies some suitable extra conditions (see (1.2), (1.3)). However, with the exception of the recent papers [BV4], [BV5] (see also [K], [Be]) not so much attention has been payed to "alternative natural assumptions" that p may fulfill, in order that (v, p) be regular. By "alternative natural assumptions", we mean assumptions that formally follow from the Poisson equation relating pressure and velocity (see (1.4)). The objective of this paper is to prove that (v, p) is regular if vertical bar p vertical bar/(1 + vertical bar v vertical bar) obeys some conditions that are in formal agreement with this relation.