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

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, $\mathit{u}\cdot \mathrm{\nabla }E=0,$ where $E=\frac{1}{2}\mathrm{\mid }\mathit{u}{\mathrm{\mid }}^2+p$ is the specific mechanical energy, which characterizes the transition from laminar to turbulent flow.