An H^1 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 our previous work in Re.Math.Phys. 2008. 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, the (mild) solution of the NS Cauchy problem is global for each H^1 initial datum w with zero mean, such that curl w <= 0.407; this improves the bound for global existence curl w <= 0.00724, derived recently by Robinson and Sadowski (2008) . We announce some future applications, based again on the H^1 framework and on the general scheme presented in our previous paper in Rev.Math.Phys. 2008.