  We consider the incompressible Navier-Stokes (NS) equations on a torus, in
the setting of the spaces L^2 and H^1; our approach is based on a general
framework for semi- or quasi-linear parabolic equations proposed in the
previous work [9]. We present some estimates on the linear semigroup generated
by the Laplacian and on the quadratic NS nonlinearity; these are fully
quantitative, i.e., all the constants appearing therein are given explicitly.
As an application we show that, on a three dimensional torus T^3, the (mild)
solution of the NS Cauchy problem is global for each H^1 initial datum u_0 with
zero mean, such that || curl u_0 ||_{L^2} <= 0.407; this improves the bound for
global existence || curl u_0 ||_{L^2} <= 0.00724, derived recently by Robinson
and Sadowski [10]. We announce some future applications, based again on the H^1
framework and on the general scheme of [9].
