An H1H1 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 L2L2 and H1H1; 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 T3, the (mild) solution of the NS Cauchy problem is global for each H1H1 initial datum u0u0 with zero mean, such that ‖curlu0‖L2⩽0.407; this improves the bound for global existence ‖curlu0‖L2⩽0.00724, derived recently by Robinson and Sadowski (2008) [3]. We announce some future applications, based again on the H1H1 framework and on the general scheme of [5].