Global regularity for some classes of large solutions to the Navier-Stokes equations

TL;DR

This paper extends previous work on the Navier-Stokes equations by establishing global regularity results for large initial data that vary slowly or are ill prepared, using the structure of the nonlinear term.

Contribution

It generalizes earlier results to include ill prepared initial data with unbounded norms, demonstrating global regularity under broader conditions.

Findings

Global regularity for large, slowly varying initial data.

Extension to ill prepared data with unbounded norms.

Utilization of nonlinear term structure in proof.

Abstract

In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared'' (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared'' situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation.