On Geometric Evolution and Microlocal Regularity of the Navier-Stokes Equations

TL;DR

This paper develops a microlocal geometric framework for analyzing the regularity and evolution of solutions to the 3D incompressible Navier-Stokes equations on Riemannian manifolds, linking geometric controls to solution extendibility.

Contribution

It introduces a novel microlocal-Riemannian approach that characterizes solution regularity via intrinsic geometric and microlocal controls, offering new insights into Navier-Stokes regularity issues.

Findings

A microlocal energy and entropy are defined and analyzed for the flow.

Smooth solutions fail to extend if specific microlocal controls break down.

On the flat torus, all assumptions are verified explicitly.

Abstract

We propose a microlocal-Riemannian framework for the three-dimensional incompressible Navier-Stokes equations on a smooth oriented Riemannian manifold (M,g). The dynamics is lifted to the unit cosphere bundle S*M via a normal-coordinate microlocal transform whose construction is justified by the positive homogeneity of the principal symbol of the linearised system in the cotangent fiber variable. Once the velocity field is fixed, the lifted dynamics is a linear non-autonomous transport-dissipation equation on a compact phase space; its coefficients encode intrinsic geometric quantities of the original flow. We introduce a microlocal energy, an angular volume functional and a directional entropy, and analyse their dissipation along the lifted dynamics. An effective affine connection encodes the back-reaction of the velocity gradient on the geometry of S*M and gives rise to a Ricci-type microlocal evolution. The framework yields a sharp geometric equivalence: a smooth solution fails to extend past time T if and only if at least one of three intrinsic microlocal controls -- deformation integrability, directional-entropy boundedness, or lifted-energy boundedness -- fails. A dimensional analysis exhibits a symmetry-lock phenomenon, in which the asymptotic vanishing of the volume of fibers enforces angular isotropy and topologically obstructs the formation of microlocal angular singularities. The framework is illustrated explicitly on the flat torus, where every assumption is verified, and is extended to the Euclidean setting under a uniform-coordinate hypothesis. The global regularity problem is not resolved here; rather, it is recast as a structural-stability question for a compact, symmetry-constrained, microlocally coercive evolution system, with the role of viscosity made explicit through spectral coercivity.