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

TL;DR

This paper develops quantitative estimates for the Navier-Stokes equations in H^1 spaces on a torus, providing explicit bounds for global existence of solutions and improving previous results.

Contribution

It introduces a new H^1 framework for Navier-Stokes, with explicit estimates, leading to improved bounds for global solutions on the torus.

Findings

Explicit constants for linear semigroup estimates

Global existence for initial data with curl u_0 in L^2 up to 0.407

Improved bound over previous results (0.00724 to 0.407)

Abstract

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].