There is a buzz in the Philippine science circle that Amador Muriel has solved the 3D Navier-Stokes Equation. This equation (or an understanding of this equation) is one of the Millenium Prize Problems of the Clay Mathematics Institute (CMI). These problems collated by CMI are “some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium.”
Honestly, I am a skeptic but I also could not comment intelligently on the merits of the work as it is outside my field of expertise. And so I thought of blogging this to get feedbacks.
A formal presentation of the problem is written by Charles Fefferman (click here). The CMI asks for a proof of one of the following:
(A) Existence and smoothness of Navier–Stokes solutions on R3.
(B) Existence and smoothness of Navier–Stokes solutions in R3/Z3.
(C) Breakdown of Navier–Stokes solutions on R3.
(D) Breakdown of Navier–Stokes Solutions on R3/Z3.
Muriel in his arxiv paper  writes that he has arrived at an exact solution of this well-known problem by continuing on their work from Physica D (1997) .
In this paper, they use a molecular picture in their treatment of hydrodynamics. Hence, giving up on the continuum model. Interestingly save for some self-citations, the paper was unheralded.
The claim in the arxiv paper (which claims to have solved the NSE) are as follows:
- They have time evolved expressions that yield velocity fields, energy fields and pressure.
- The self-consistent non-linear solutions they have, are all continuous fields and there are no blowup times.
- There is no turbulence in the NSE.
They said however, that they did not directly attack the NSE problem but attacked it from “direct contracted results from the Liouville equation”. But they made a comparison between the solution for NSE and their time evolved equations and they said that they are “qualitatively the same”.
So what do you think?
 Muriel, A. (2011). An Exact Solution of the 3-D Navier-Stokes Equation arXiv:1011.6630v1 [math-ph]
 Muriel, A. (1997). An integral formulation of hydrodynamics Physica D: Nonlinear Phenomena, 101 (3-4), 299-316 DOI: 10.1016/S0167-2789(96)00181-9