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 the previous work (Morosi and Pizzocchero, Rev. Math. Phys. 20 (2008) 625–706). 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 u0 with zero mean, such that ‖curl u0‖_{L^2} ⩽ 0.407; this improves the bound for global existence ‖curl u0‖_{L^2} ⩽ 0.00724, derived recently by Robinson and Sadowski (Asymptot. Anal. 59 (2008) 39–50). We announce some future applications, based again on the H^1 framework and on the general scheme of (Morosi and Pizzocchero, Rev. Math. Phys. 20 (2008) 625–706).