Large, global solutions to the Navier-Stokes equations, slowly varying in one direction

TL;DR

This paper constructs new large initial data examples for the 3D Navier-Stokes equations, which vary slowly in one direction and still produce global smooth solutions, expanding understanding of solution existence.

Contribution

It introduces a novel class of large initial data with slow variation in one direction that guarantees global solutions, differing from previous oscillatory data examples.

Findings

Existence of global solutions for large, slowly varying initial data.

New examples of initial data with no oscillatory properties.

Utilization of the nonlinear term's structure in the proof.

Abstract

In to previous papers by the 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 aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions, in the whole space. Contrary to the previous examples, the initial data has no particular oscillatory properties, but varies slowly in one direction. The proof uses the special structure of the nonlinear term of the equation.