2009/09/21 by Carlo Morosi, Livio Pizzocchero · 6 citations
Mathematics · #math.AP #msc:35Q30 #msc:76D03 #msc:76D05
paper · pdf · doi:10.1016/j.na.2010.11.043
published as Nonlinear Analysis: TMA 74(6) (2011), 2398-2414 (in a slightly revised version) · LaTeX; 33 pages
arxiv created 2009/09/21 · arxiv updated 2011/02/18
We consider the incompressible Navier-Stokes (NS) equations on a torus, in the setting of the spaces L2 and H1; our approach is based on a general framework for semi- or quasi-linear parabolic equations proposed in the previous work [9]. 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 H1 initial datum u0 with zero mean, such that || curl u0 ||L2 <= 0.407; this improves the bound for global existence || curl u0 ||L2 <= 0.00724, derived recently by Robinson and Sadowski [10]. We announce some future applications, based again on the H1 framework and on the general scheme of [9].