Self-similar singularity profiles for two-dimensional Boussinesq
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Which self-similar profiles can describe blow-up in the two-dimensional Boussinesq equations, and what are their linearized unstable modes? Stable and unstable profiles are separate targets; numerically small residuals provide candidates rather than a rigorous existence theorem.
Why it matters and what counts as progress
Why it matters: Unstable self-similar profiles reveal additional candidate blow-up branches not covered by proofs for related stable profiles.
What counts as progress: Find additional profiles with controlled residuals and stability calculations, or prove that a numerical candidate corresponds to an exact solution.
How the problem developed
- 17 Sept 2025 · Numerical discovery [1]
Yongji Wang and coauthors Reported one stable and three unstable self-similar Boussinesq profiles, with residuals of orders 10⁻⁸ to 10⁻⁷; a fourth unstable profile remained an unvalidated candidate.
AI contribution
Discovery of Unstable Singularities
Wang and collaborators report a second unstable self-similar Córdoba–Córdoba–Fontelos (CCF) profile, stable and three unstable incompressible porous media (IPM) profiles, and stable and three unstable 2D Boussinesq profiles. They test residuals and linearized modes on dense grids. A fourth Boussinesq profile remains an unvalidated candidate.
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Publication dates and editorial changes.
Neural networks find numerical candidates for unstable fluid singularities