On the global wellposedness of the 3-D Navier-Stokes equations with large initial data

TL;DR

This paper establishes a new nonlinear condition under which the 3-D Navier-Stokes equations are globally well-posed, allowing for arbitrarily large initial data in a critical function space.

Contribution

It introduces a nonlinear criterion for global well-posedness that does not require small initial data, expanding the class of initial conditions ensuring smooth solutions.

Findings

Global well-posedness under nonlinear initial data condition

Construction of explicit large initial data leading to smooth solutions

Extension of known global existence results to larger data sets

Abstract

We give a condition for the periodic, three dimensional, incompressible Navier-Stokes equations to be globally wellposed. This condition is not a smallness condition on the initial data, as the data is allowed to be arbitrarily large in the scale invariant space $ B^{-1}\_{\infty,\infty}$, which contains all the known spaces in which there is a global solution for small data. The smallness condition is rather a nonlinear type condition on the initial data; an explicit example of such initial data is constructed, which is arbitrarily large and yet gives rise to a global, smooth solution.