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

TL;DR

The paper tackles the 3D Navier–Stokes regularity problem by embedding the dynamics into a covariant geometric framework and lifting the system to the cosphere bundle $S^*M$, where velocity and vorticity are represented as microlocal distributions. It develops a transport–dissipation model driven by the lifted flow $X_u$, augmented by an effective connection and a Ricci-type geometric evolution through a deformed metric $g'$; viscosity induces geometric diffusion and provides monotone control of microlocal invariants. Core contributions include the introduction of microlocal amplitudes, directional energy functionals $\,\mathcal{V}_\lambda$, a vorticity functional $W[\tilde{\omega}]$, and a Green operator $G(t,s)$ for explicit solution representation, yielding geometric blow-up criteria in terms of directional concentration, vortex-stretching alignment, and temporal integrability of $\|\nabla u\|_{L^\infty}$. The framework reframes regularity as dissipative stability on a compact phase space, offering a principled geometric mechanism to constrain blow-up scenarios and guiding future structure-preserving numerical methods, even though a full resolution of global regularity remains open.

Abstract

We develop a geometric and microlocal framework for the Navier--Stokes equations by lifting the dynamics to the cosphere bundle of a Riemannian manifold. In this formulation, the velocity field and vorticity are represented as microlocal distributions whose evolution is governed by a transport--dissipation system generated by a canonical dynamical vector field described with differential forms. We introduce microlocal amplitudes, directional energy functionals, and monotone volume invariants on the compact phase space, which quantify directional concentration and alignment mechanisms associated with potential loss of regularity. The viscous term induces effective geometric diffusion on the cosphere bundle, yielding closed differential inequalities in a geometric setting. To capture the interaction between fluid deformation and geometry, we define an effective connection and curvature tensor encoding the influence of the symmetric velocity gradient. This structure gives rise to a Ricci-type geometric evolution at the microlocal level, controlled by a dissipative functional that excludes extreme directional concentration and persistent transverse stretching. Our results do not resolve the global regularity problem in full generality, but provide a coherent geometric mechanism that severely restricts blow-up scenarios compatible with viscous Navier--Stokes dynamics, reformulating the analytic regularity question as a problem of dissipative stability on a compact phase space.