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

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

2010-02-25

arxiv.org/abs/1002.4736v2

Mathematics

Analysis of PDEs

Accepted for publication in Journal f\"ur die reine und angewandte Mathematik

Scientific paper

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.

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