Self-similar singularity profiles for CCF
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Which self-similar blow-up profiles exist for the Córdoba–Córdoba–Fontelos equation, and how many unstable modes do they have? A stable profile and the first unstable profile were known before searches for further branches.
Why it matters and what counts as progress
Why it matters: Additional unstable CCF profiles extend the blow-up landscape beyond the stable branch and first unstable solution.
What counts as progress: Find a new unstable branch with controlled numerical residuals and modes, or prove existence and instability of a computed profile.
How the problem developed
- 2023 · Numerical candidate [1]
Jiajie Wang, Ke Chen, Thomas Y. Hou and Zhen Lei Reported numerical discovery of the first unstable self-similar profile for the Córdoba–Córdoba–Fontelos equation, alongside stable profiles for related fluid models.
- 17 Sept 2025 · Numerical candidates [2]
Jiajie Wang and collaborators Reported a second unstable CCF profile and multiple stable or unstable profiles for IPM and 2D Boussinesq equations, with dense-grid residual and linearized-mode tests.
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.
Sources
Website timeline
Publication dates and editorial changes.
Neural networks find numerical candidates for unstable fluid singularities