Self-similar singularity profiles for incompressible porous media
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For the incompressible porous media (IPM) equation, can self-similar profiles describe finite-time singular behavior, and which profiles have stable or unstable linearized modes? The focus is on identifying profiles and their instability structure, with numerical candidates distinguished from exact solutions.
Why it matters and what counts as progress
Why it matters: Self-similar profiles offer possible mechanisms for singularity formation in the IPM model. Their unstable modes help distinguish candidate families beyond the stable profiles found earlier.
What counts as progress: Find additional profiles, improve controlled residual and mode estimates, or prove that an exact profile and the corresponding singular solution exist. Dense finite-grid tests alone do not establish existence. Results for related CCF or Boussinesq equations do not automatically transfer to IPM or Navier–Stokes.
How the problem developed
- 17 Sept 2025 · Numerical discovery [1]
Yongji Wang and coauthors Reported a stable and three unstable self-similar IPM profiles, with equation residuals of orders 10⁻¹¹ to 10⁻⁸, found using neural-network search and high-precision optimization.
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