Episode 73 · The Machine's Theorems, part 5 of 20 (Arc 2: Navier–Stokes)

The Million-Dollar Question: A Primer on Regularity

Sankalp Thakur · 50 minutes

Farah watches her daughter spin on the ice and asks what stops a fluid's vortex tubes from spinning up forever. The episode covers the vorticity equation (stretching, carrying, diffusing) and why stretching vanishes in 2D; axisymmetric flows with and without swirl; the cascade as vortex stretching; the Burgers vortex balance; and Leray's lower bound on blowup speed. Beale-Kato-Majda turns blowup into one quantity to watch. Caffarelli-Kohn-Nirenberg partial regularity, which Fefferman calls the best known, allows isolated points but no moving singularities. ESS closes the L-three endpoint using backward uniqueness. Convex integration runs from Nash-Kuiper through De Lellis-Székelyhidi and Isett to Buckmaster-Vicol. Albritton-Brué-Colombo show two distinct suitable Leray-Hopf solutions from rest under the same (rough) force. Tao's averaged equation sets the supercriticality barrier and inspired the fluid-computer dream. The episode also covers hyper- and hypo-dissipation and Lions's five-quarters threshold, the forcing trick ("the hard part is making the leftover smooth"), and a seven-question checklist for any blowup claim. Family 376 is mentioned as an unrefereed claim. Farah Al-Mansoori and Augustin Lévêque are fictional characters. All quotations from real people come from the sources listed in the verification ledger.

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