Rigorous unstable self-similar singularities in incompressible fluid models
In plain words
Computer searches in 2025 found self-similar blowup solutions that occur only for finely tuned starting flows (unstable singularities) in simpler 2D fluid models. Turning these numerical profiles into proofs is open.
Precise statement
Wang et al. (arXiv 2509.14185, 2025) computed families of unstable self-similar profiles to high precision for the Cordoba-Cordoba-Fontelos model, the 2D incompressible porous media equation and the 2D Boussinesq equations with boundary. Prove that at least one such profile yields an exact finite-time singularity from smooth data, by a computer-assisted stability estimate on the finite-codimension set of data attracted to it. Answer: a proof for each profile family.
What would settle it
A computer-assisted proof of nonlinear stability modulo finitely many unstable directions for one profile of the unforced equations.
Status in the literature
Unverified note
Profiles were computed numerically in 2025; a rigorous stability proof was the stated next step.