An H^1 setting for the Navier-Stokes equations: quantitative estimates

Details An H^1 setting for the Navier-Stokes equations: quantitative estimates An H^1 setting for the Navier-Stokes equations: quantitative estimates

2009-09-21

arxiv.org/abs/0909.3707v1

Nonlinear Analysis: TMA 74(6) (2011), 2398-2414 (in a slightly revised version)

Mathematics

Analysis of PDEs

LaTeX; 33 pages

Scientific paper

10.1016/j.na.2010.11.043

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- 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 T^3, the (mild) solution of the NS Cauchy problem is global for each H^1 initial datum u_0 with zero mean, such that || curl u_0 ||_{L^2} <= 0.407; this improves the bound for global existence || curl u_0 ||_{L^2} <= 0.00724, derived recently by Robinson and Sadowski [10]. We announce some future applications, based again on the H^1 framework and on the general scheme of [9].

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