What question did this study set out to answer?

This work aims to establish a BV chain rule for log-Lipschitz projections in dissipative dynamical systems.

May 16, 2026Open Access

Log-Distance Integrability for Dissipative Flows: BV Selections on Fractal Attractors

Key Points

This work aims to establish a BV chain rule for log-Lipschitz projections in dissipative dynamical systems.
Development of conditions for log-distance integrability along trajectories.
Identification of sufficient routes to integrability, including bounded variation and power-law control.
Numerical verification of the framework on chaotic attractors like the Lorenz attractor.
Conditions for time derivative of log-distance functional maintain integrability under specific trajectories.
Major corrections have clarified LDI's power-law sublevel-time control structure.
First numerical verification shows success across multiple tests on Lorenz attractor, validating the established framework.

Abstract

We establish a BV chain rule for log-Lipschitz nearest-point projections onto compact attractors of dissipative dynamical systems. The main result provides conditions under which the time derivative of the log-distance functional d (u (t), A) remains integrable along trajectories, yielding BV regularity of the projected trajectory. DRAFT version. Subject to revisions. The Navier–Stokes transfer remains conditional and should be read as a selection-regularity framework, not as an unconditional 3D Navier–Stokes regularity theorem. Two sufficient routes to LDI are identified: bounded variation of the log-distance and power-law sublevel-time control. The v1. 4 revision adds guardrails for strong versus trajectory-attractor formulations, no-positive-measure contact, nonconvex regularised selectors, and skeleton compactness. CHANGELOG Changes in Version 1. 4 (May 2026) Critical: Guardrails added for the 3D Navier–Stokes attractor status: the framework now distinguishes strong single-valued semiflows from weak/trajectory-attractor formulations. Major: LDI Variant B clarified as power-law sublevel-time control; no-positive-measure contact is no longer treated as an automatic Leray–Hopf consequence. Critical: The LDI tail-integral equivalence was corrected to include the speed weight ||u' (t) || log (R/d (t) ) ; the unweighted int log (1/d (t) ) dt condition is no longer used as the LDI formula. Major: The time-averaged selection and damped-to-classical transfer are recast through pre-registered selectors and a skeleton compactness criterion. Changes in Version 1. 3 (March 2026) Major: PROOF OF LIFE — TLL+LDI numerically verified on the Lorenz attractor (all 5 tests passed: CTLL=0. 19, LDI finite, BV chain rule ratio=0. 52, STC α=1. 11, exponential tracking λ=0. 064) Major: First numerical verification of the TLL+LDI framework on a non-trivial chaotic attractor Changes in Version 1. 2 (March 2026) Critical: Fixed thanks placement

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