An H1 setting for the Navier-Stokes equations: Quantitative estimates

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-linear or quasi-linear parabolic equations proposed in the previous work (Morosi and Pizzocchero (2008) [5]). 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 parallel to curl u(0)parallel to(L2) <= 0.407; this improves the bound for global existence parallel to curl u(0)parallel to(L2) <= 0.00724, derived recently by Robinson and Sadowski (2008) [3]. We announce some future applications, based again on the H-1 framework and on the general scheme of [5]. (C) 2010 Elsevier Ltd. All rights reserved.