An Evolution Criterion for the 3D Navier–Stokes Equations Based on Critical Energy

This paper presents a complete framework for describing the global evolution of solutions to the 3D incompressible Navier-Stokes equations. There exists a critical kinetic energy Ec such that if the initial kinetic energy is below Ec, the solution is globally smooth and remains regular forever; if it is above Ec, the solution blows up in finite time, corresponding to turbulence and singularity formation. Due to energy dissipation, all solutions eventually tend to a low-energy state and recover smoothness. Only the core framework and main conclusions are presented in this paper. The full rigorous mathematical proof will be published in future work.