On Leray’s blow-up condition

In his ground breaking paper [4], Leray proves that if the velocity [Formula: see text] is a smooth solution to the 3-dimensional, incompressible, Navier–Stokes equations, that blows up in finite time [Formula: see text], then [Formula: see text]. This provides a lower bound for possible blowouts of solutions to the Navier–Stokes equations. The possibility of a blowout occurring at this minimal level was explored in [3,5]. This paper supplements and extends these results, looking in more detail at the concept of ancient solutions.