x.com/BetterCallMedhi/status/2097472336257863722
1 correction found
the clay mathematics institute millennium prize does not ask m whether you can artificially force a singularity in a fluid by injecting an ad hoc smooth external forcing term f(x,t) to twist the vortex until it breaks the real problem questions the fundamental stability and global smooth existence for 3dimensional incompressible euler & navier stokes equations under natural conservation laws and viscous dissipation alone
This misstates the official Clay problem. The Clay Millennium Prize formulation explicitly allows a smooth external force in its breakdown alternatives, and Euler is not itself one of Clay’s Millennium Prize problems.
Full reasoning
The official Clay Mathematics Institute problem statement by Charles Fefferman does not limit the prize problem to unforced Navier–Stokes or to Euler-plus-Navier–Stokes together.
In Fefferman’s statement, Clay says solvers may prove one of four alternatives. The two breakdown alternatives, (C) and (D), explicitly allow a smooth external force f(x,t):
- (C) asks for existence of a smooth divergence-free initial field and a smooth
f(x,t)onR^3 × [0,∞)such that there are no global smooth solutions satisfying the listed conditions. - (D) asks the analogous periodic version, again with a smooth
f(x,t).
So the post’s claim that Clay “does not ask” whether singularity can occur with a smooth external forcing term is directly contradicted by the official problem description.
The post also incorrectly folds Euler into the Millennium Prize statement. Fefferman writes that the analogous Euler questions are “open and very important,” but Euler is not on Clay’s list of prize problems. The Millennium Prize problem here is specifically Navier–Stokes existence and smoothness.
OpenAI’s announcement itself says its claimed result resolves statement “C” (and also “D”) of the official Clay formulation, i.e. the forced-breakdown branches that Clay explicitly included.
3 sources
- Existence and Smoothness of the Navier–Stokes Equation — Charles L. Fefferman / Clay Mathematics Institute
“We ask for a proof of one of the following four statements.” Statement (C): “there exist a smooth, divergence-free vector field ... and a smooth f(x,t) ... for which there exist no solutions ...” Statement (D) likewise includes “a smooth f(x,t)” in the periodic case.
- Existence and Smoothness of the Navier–Stokes Equation — Charles L. Fefferman / Clay Mathematics Institute
“These problems are also open and very important for the Euler equations (ν = 0), although the Euler equation is not on the Clay Institute’s list of prize problems.”
- On the Navier–Stokes Millennium Prize Problem | OpenAI
OpenAI writes: “This resolves the Navier–Stokes Millennium Prize problem by establishing statement ‘C’ (and also ‘D’) in the official Millennium Prize formulation.”