Did Open AI's Collective Of Agents Really Solve The Navier -Stokes Equations (In 3D)? Not Quite

AMS Newsroom

Statement from American Mathematical Society Leadership on Navier-Stokes Problem:

"The news today of progress on resolving the Navier–Stokes problem, one of mathematics' great longstanding challenges concerning the equations that govern the flow of fluids, represents a milestone advance in human knowledge. This story began with Navier, Stokes, Leray, and Ladyzhenskaya and has culminated in the recent breakthroughs of Córdoba and Martínez-Zoroa, then — assisted by new technologies — Alpöge and Buckmaster, with the final steps taken by OpenAI mathematicians. The purpose of mathematics is human understanding, and this achievement, and the process that led to it, will bear fruit for a long time to come."

The statement by the AMS does not assert a solution has been found to the 3D Navier-Stokes equation, mainly because there is still an open debate. (See Note 1).   This is mainly because the Open A.I. chosen "forced fluid scenario" relies on a contrived external force that does not occur in real world fluid dynamics. Removing that contrived force causes the mathematical singularity ('blow up') to disappear.  This is neat, but then little useful insight emerges into actual physical turbulence (or the unforced Navier-Stokes problem researchers really care about.)

Despite this, it is still possible to approach to the history and background of the Navier-Stokes equation, to puts in perspective this partial achievement, starting with the classic 2D methods.  To fix ideas we are concerned with an 'ideal fluid' moving under an internal force known as the pressure gradient, i.e.  Ñ p. 

The original Euler formulation in 1755 recast Newton's 2nd law (F = m dv/dt) in terms of the velocity vector field  v (x (t), t) and the scalar pressure p (x (t) t) of an incompressible fluid.  Then the form of the resulting equations is:

1) ¶ v/ ¶ t + (v  · Ñ) v  =    - Ñ p   (x Î R ⁿ ,  t > 0}  

1a)  Ñ · v  = 0  

With the initial condition: v (x (t) = v_o (x)

Where  v_o (x)  is the given divergence-free vector field.  Jump forward now to  1822 and the French mathematician Navier has derived a modification of the Euler equations, viz.

2) ¶ v/ ¶ t + (v  · Ñ) v  =  e D v  - Ñ p   (x Î R ⁿ ,  t > 0}  

And:   Ñ · v  = 0  

Again with the same initial condition.  Note that effects of non-linearity are "strikingly different" in two vs. three dimensions.   The 2D Navier-Stokes equations are classical results proved in 1933 (by Jean Leray) while the 3D version amounts to the same Clay prize problem. (Ref. 1).

Navier included the factor  e as simply a function of the molecular spacing. Also a crucial difference between the 2D and 3D scenarios is the constraint that eqns. (1) and (2) impose (in 2D) on the evolution of the vorticity. (op. cit.) Taking the curl of (1) and (2) gives the equations for the evolution of the vorticity in an inviscid fluid, the first of which:

¶w/ ¶ t + (v  · Ñ)w   =    (w  · Ñ) v

I noted (in my Aug. 11 post) this equation could be applied to the newly discovered solar vortices, e.g.

given that the solar plasma is highly inviscid. But in the curls noted one other result appears for a viscous fluid:

¶w/ ¶ t + (v  · Ñ)w   =    (w  · Ñ) v  +    e D w

which can be seen as the analog to Navier's modification of the classical Euler equations in 2D. (Equations (2) above). Note the term (w · Ñ) v  vanishes in 2D. Still, although the inviscid and viscous equations remain nonlinear they are significantly simpler than the 3D equations. (Op. cit., Ref. 1)  In 2D the inviscid equation (the one which also may apply to solar vortices) becomes:

¶w/ ¶ t + (v  · Ñ)w   =    d w/ d t  = 0

So in 2D the vorticity is a scalar conserved along the trajectories of the fluid particles. (Ibid.)  Interestingly, where the solar vortices are concerned it is precisely the Kelvin circulation (or Kelvin-Helmholtz) instability where the relevant theorem for 3D flows preserves the (w  · Ñ) v  term so there is no vanishing.  This "may be instrumental in creating blowup for the Euler equations". (Ref. 1, p. 9) 

 All of this shows the solution to the Navier-Stokes equations may be a bit farther away than the Open AI agents solutions suggest. In any case nothing will be final until the results get published in a peer-reviewed paper. (And claims are settled with mathematicians who insist they got there first.)

Ref. 1:Cannone, Marco and Friedlander, Susan: Notices of the AMS, Navier: Blowup and Collapse, January, 2003, p. 7.

 Note 1: The Clay Mathematics Institute still lists the problem as unsolved. Official validation requires formal publication in a peer-reviewed journal and a multi-year community review process, and OpenAI stated it does not plan to claim the $1 million prize.(In September, Open AI announced that a collective of coordinating agents used an unreleased model to produce a proof showing the equations can develop a singularity or blowup in finite time.