Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries
Bibliographic record
Abstract
A family of explicit solutions is described, to the porous medium equation in its full range of nonlinearities (plus some analogous fourth-order diffusions), in which the pressure is given by a quadratic function of space at each instant in time. These include spreading solutions whose source is concentrated on any conic region of dimension lower than the ambient space, and solutions which focus at conic regions. The singular limiting distributions are affine projections of Barenblatt type solutions (with arbitrary signature) onto lower dimensional subspaces. All affine images of Barenblatt solutions form an invariant space on which the dynamics can be integrated explicitly. A time-reversal symmetry is revealed for the pressure equation which transforms spreading solutions to focusing solutions, and vice-versa. This yields new information about the long and short time asymptotics of finite-mass solutions, about the instability of focusing, and about singularity geometry. Résumé On décrit une famille de solutions explicites pour l'équation des milieux poreux pour toute l'échelle des nonlinéarités (ainsi que pour d'autres équations de diffusion analogues de quatrième degré). Pour ces solutions, la pression à chaque instant est une fonction quadratique en \boldsymbol x . Certaines de ces solutions sont diffusées à partir d'une source concentrée sur un domaine conique arbitraire de dimension inférieure à celle de l'espace ambient ; d'autres se focalisent sur des régions coniques. Pour les solutions focalisantes, on obtient à la limite une distribution singulière qui est une projection affine d'une solution de type Barenblatt (à signature arbitraire) sur un espace de dimension inférieure. L'ensemble des images affines des solutions de type Barenblatt est un espace invariant, sur lequel la dynamique peut s'intégrer explicitement. Par un changement de variables qui inverse le sens du temps, on arrive à transformer une solution diffusive en une solution focalisante. Ceci donne de nouvelles informations sur l'asymptotique en temps court et long pour des solutions de masse finie, sur l'instabilité des solutions focalisantes, ainsi que sur la géométrie des singularités.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".