Post by @tao@mathstodon.xyz

2026-09-03 15:37 UTC

But attempts to resolve either of these two questions have historically led to fundamental insights and influential theorems in fluid mechanics, analysis, and partial differential equations: the Leray-Hopf weak solutions, the Gagliardo-Nirenberg-Ladyshenskaya inequalities, the Prodi-Serrin partial regularity theorems, the Beale-Kato-Majda blowup criterion, the Escuriaza-Seregin-Sverak conditional regularity result, and so forth. The broader theory of turbulence, while not directly connected to any of these results, has certainly been informed at a philosophical level at least by the efforts to establish or disprove global regularity. One of my own contributions to the subject was to introduce the concept of fluid computation and Turing universality to the problem, which among other things led to the unexpected connections with symplectic topology. (2/6)

Replies (1)

@tao@mathstodon.xyz 2026-09-03 15:38

With recent advances in understanding related fluid equations, it is now the emerging consensus that the answer to the global regularity problem for Navier-Stokes is negative: there should exist very specific initial conditions to this problem that develop singularities in finite time. There is even a reasonably well-defined strategy to locate such conditions: (a) Design a nearly-self-similar ansatz for a finite time blowup solution. (b) Locate an approximate solution to this ansatz, which numerically obeys the ansatz up to an extremely small, computable residual. (c) Demonstrate that, in suitably renormalized coordinates, the ansatz is stable around this numerical solution, and can be perturbed into an exact solution if the residual is small enough. (d) Verify that the residual of the solution falls within the threshold of stability of the solution. (3/6)

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