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 View the MathML sourceT3, the (mild) solution of the NS Cauchy problem is global for each H1H1 initial datum u0u0 with zero mean, such that View the MathML source‖curlu0‖L2⩽0.407; this improves the bound for global existence View the MathML source‖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].