DOI: 10.5281/zenodo.18304313
title: A Lyapunov-Perelman Accounting Resolution of the Navier-Stokes Regularity Problem

description: A Lyapunov-Perelman Accounting Resolution of the Navier-Stokes Regularity Problem

The three-dimensional incompressible Navier–Stokes regularity problem turns on whether nonlinear vortex stretching can generate a finite-time singularity from finite-energy initial data. While intense local amplification is ubiquitous in turbulent flows, a genuine blow-up would require sustained, non-integrable accumulation of curvature. In this work, we show that this accumulation mechanism is structurally unavailable within the physically admissible Leray–Hopf class. Rather than seeking stronger a priori bounds or conditional continuation criteria, we isolate the unique structural channel through which singularity could occur and demonstrate that the Navier–Stokes dynamics themselves prohibit it.

Our approach reformulates vortex stretching as a curvature-debt accounting problem. Beginning with the canonical advection–diffusion density induced by a Leray–Hopf velocity field, we construct an entropy geometry intrinsic to the flow that records how curvature concentration modifies transport and dissipation. This geometry induces a self-adjoint, semi-bounded operator whose quadratic form encodes confinement and repayment responses to localization. Crucially, this framework is diagnostic rather than restrictive: it introduces no auxiliary smoothness, integrability, probabilistic closure, or admissibility assumptions beyond the Leray–Hopf formulation posed by the Clay problem.

Within this setting, we derive a Lyapunov–Perelman–type accounting law that balances instantaneous curvature amplification against entropy-driven confinement and viscous dissipation. The balance has a sign-definite structure: while localized amplification events are permitted, persistent curvature debt is dynamically forbidden. We prove that the amplification functional governing vortex stretching is integrable on any finite time interval at Leray–Hopf regularity, precluding the non-integrable accumulation required for finite-time blow-up. This closes, rather than refines, the classical blow-up narrative by eliminating its sole structural trigger.

To ensure mathematical legitimacy at weak regularity, the time-dependent operator lift is placed within the Friedrichs–Kato theory of measurable quadratic forms, yielding a well-defined resolvent calculus and rigorous weak evolution identities. No hidden regularity enters the argument: all quantities are canonically induced by the flow and remain meaningful at the exact level of regularity demanded by the Clay Millennium statement. As a result, the exclusion of finite-time singularities follows directly from the intrinsic structure of the Navier–Stokes equations themselves.

A central contribution of this work is the conceptual shift from instantaneous amplification to cumulative accounting. Classical criteria such as Beale–Kato–Majda and Serrin–Prodi identify quantities whose divergence would signal breakdown but do not determine whether such divergence is dynamically admissible. Here we show that the only quantity capable of driving curvature growth—the stretching amplification functional—is automatically integrable in time for Leray–Hopf solutions. Consequently, all known blow-up scenarios share a necessary condition that the equations forbid.

The theory is rendered falsifiable through empirical validation. We demonstrate that the same accounting structure is realized in strongly driven Weber packed-bed transport experiments and in high-Reynolds-number channel-flow simulations from the Johns Hopkins Turbulence Database. Across both laboratory and DNS regimes, amplification, pressure redistribution, spectral transfer, and wall-stress observables remain bounded, localized, and temporally stationary. No evidence of cumulative curvature debt or runaway amplification is observed, even under sustained forcing, directly mirroring the theoretical accounting law.

Taken together, the analytical and empirical results indicate that Navier–Stokes dynamics enforce an intrinsic regularity mechanism: the equations permit borrowing through local amplification, but they enforce repayment through entropy-induced geometric confinement and dissipation. Finite-time blow-up would require a violation of this accounting principle, and no such violation is admissible within the Leray–Hopf class. In this sense, the work resolves the Navier–Stokes regularity problem by uncovering—and empirically corroborating; a fundamental structural law already encoded in the equations themselves.