Sums of large global solutions to the incompressible Navier-Stokes equations

TL;DR

This paper demonstrates that large global solutions to the 3D incompressible Navier-Stokes equations can be perturbed by slowly varying large fields, revealing an uncountably infinite structure of such solutions.

Contribution

It shows that solutions in the set G are stable under large, slow perturbations, indicating a rich, uncountably infinite structure of global solutions.

Findings

Any element of G can be perturbed by a large, slowly varying field to produce another element of G.

Uncountably many long segments of solutions in G pass through any given solution.

The result highlights the complex structure of the solution space for the Navier-Stokes equations.

Abstract

Let G be the (open) set of~$\dot H^{\frac 1 2}$ divergence free vector fields generating a global smooth solution to the three dimensional incompressible Navier-Stokes equations. We prove that any element of G can be perturbed by an arbitrarily large, smooth divergence free vector field which varies slowly in one direction, and the resulting vector field (which remains arbitrarily large) is an element of G if the variation is slow enough. This result implies that through any point in G passes an uncountable number of arbitrarily long segments included in G.