A Hartman-Grobman Theorem for the Navier-Stokes Equation on ${\mathbb{T}}^3$

Abstract

We prove a local Hartman-Grobman theorem at zero for the unforced Navier-Stokes equation on the three-dimensional torus in the strong topology of $H^s$ for every $s>7/2$. More precisely, we construct a homeomorphism between open positively invariant neighborhoods of zero that conjugates the Navier-Stokes semiflow to the heat semigroup. We develop an infinite-dimensional boundary-time scheme built on a nested hierarchy of exact asymptotic dynamical factors. Their fiber geometry yields Sobolev block coordinates with autonomous finite-dimensional factor systems, on which boundary-time topological conjugacies can be constructed at every finite level and then assembled on a common neighborhood into the desired $H^s$ conjugacy. The infinite-dimensional assembly relies on new spectral estimates for the divergence-free transport structure, which control derivative loss and the blockwise tails of the finite-level conjugacies and their inverses without an additional spectral-gap assumption.