Terence Tao Discusses Finite-Time Blowup in Three Incompressible Fluid Equations with Smooth Forcing

Summary

Terence Tao discusses recent work by Levent Alpöge and Tristan Buckmaster, building on Diego Córdoba and Luis Martínez-Zoroa, on finite-time singularity formation in incompressible fluid equations. The researchers establish blowup with smooth forcing for three simpler models: the incompressible porous medium equation, the two-dimensional Boussinesq equation, and the three-dimensional incompressible Euler equation. The work does not yet prove the corresponding result for the three-dimensional Navier-Stokes equations, but Tao says the method appears likely to extend to them and could make that goal feasible. The construction repeatedly adds small, rapidly increasing high-frequency corrections to a lower-frequency forced solution. The corrections are designed to make the solution increasingly singular while keeping the forcing term smooth; their evolution is controlled by an instability in a suitably chosen background flow. For the Boussinesq equation, an explicit near-blowup ansatz behaves linearly in space and like a high-frequency plane wave, reducing the mechanism to exactly solvable modulation ODEs. Making the construction rigorous still requires substantial technical work, including spatial cutoffs and control of nonlinear effects. The authors formalized their arguments in Lean, with heavy AI assistance, but Tao notes that the proofs were initially difficult to read and are still being rewritten into a clearer mathematical exposition. He emphasizes that understanding the ideas matters more than merely obtaining a formal result. Tao also mentions an independent preprint by Ganeshram, Duruisseaux, and Anandkumar, which uses a physics-informed neural network to locate a numerically stable candidate for an unforced Euler blowup profile. That approach has not yet established stability within the residual-error tolerance required for a rigorous proof.