To the very best of my knowledge, Kazakh professor Mukhtarbay Otelbaev submitted a paper on the solution of the Navier-Stokes regularity Millenium problem. It seems that he had found an upper bound for a unique solution for the periodic boundary condition formulation of the problem. For the readers who are familiar with the notation, the main result is stated as follows.
This result is under review as we speak. In the end, we are hoping for a solution that is constructive, i.e. an algorithm that provides the solution itself, not only shows the existence of it – if so, this work might be an important stepping stone in this process. The translation of the main result without the proof is available here. Terence Tao also has some interesting recent results.