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

Chio Chon Kit

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.

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

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, where is the specific mechanical energy, which characterizes the transition from laminar to turbulent flow.

Paper Structure

This paper contains 33 sections, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function Spaces
  4. 3D Incompressible Navier–Stokes Equations
  5. Leray Weak Solutions
  6. Derivation of the Critical Condition $\boldsymbol{u}\cdot\nabla E=0$
  7. Mechanical Energy Transport Equation
  8. Lagrangian Volume and Laminar-Turbulent Transition
  9. Rigorous Proof of the Critical Condition
  10. Weak Singularities and $H_0^1$ Regularity Collapse
  11. Critical Energy Identity
  12. Definition of Finite-Time Weak Singularities
  13. $H_0^1$ Norm Collapse
  14. Existence of Finite-Time Weak Singularities
  15. Local Well-Posedness of Strong Solutions
  16. ...and 18 more sections