Finite-Time Weak Singularities and the Statistical Structure of Turbulence in 3D Incompressible Navier-Stokes Equations

TL;DR

This paper rigorously analyzes the conditions under which smooth initial data for the 3D Navier-Stokes equations may lose regularity, focusing on the mechanical energy transport equation and identifying a critical transition condition.

Contribution

It introduces a fundamental critical condition based on the mechanical energy transport equation that characterizes the laminar-turbulent transition in 3D Navier-Stokes flows.

Findings

Derives a critical condition $oldsymbol{u} abla E = 0$ for flow transition.

Provides a rigorous mathematical framework for regularity analysis.

Focuses on mechanical energy transport rather than phenomenological models.

Abstract

This paper provides a rigorous mathematical analysis of the global regularity problem for the 3D incompressible Navier-Stokes (NS) equations, specifically addressing the conditions under which smooth initial data may lead to a loss of regularity. By departing from traditional phenomenological turbulence models and focusing strictly on the mechanical energy transport equation, we derive a fundamental critical condition, $\boldsymbol{u}\cdot\nabla E = 0,$ where $E = \frac12|\boldsymbol{u}|^2 + p$ is the specific mechanical energy, which characterizes the transition from laminar to turbulent flow.