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.
Jiawei Xie (Sun,) studied this question.