Some remarks on the regularity time of Leray solutions to the Navier-Stokes equations

In this small note we strengthen the classic result about the regularity time t* of arbitrary Leray solutions to the (incompressible) Navier-Stokes equations in Rn (n = 3, 4), which have the form: t* <= K_{3} nu^{-5} || u(.,0) ||_{L2}^{4} if n = 3, and t* <= K_{4} nu^{-3} || u(.,0) ||_{L2}^{2} if n = 4 (in particular, by reducing the current best known values for the constants K_{3}, K_{4}). Some related results of clear interest are also included (derived) in our discussion.