Sharp decay rates for the fastest conservative diffusions
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Bibliographic record
Abstract
In many diffusive settings, initial disturbances will gradually disappear and all but their crudest features — such as size and location — will eventually be forgotten. Quantifying the rate at which this information is lost is sometimes a question of central interest. Here this rate is addressed for the fastest conservative nonlinearities in the singular diffusion equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:msub> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> <mml:mo stretchy="false">/</mml:mo> <mml:mi>n</mml:mi> <mml:mo><</mml:mo> <mml:mi>m</mml:mi> <mml:mo>⩽</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">/</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="0.25em"/> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="0.25em"/> <mml:mi mathvariant="bold">x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi mathvariant="bold">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:math> which governs the decay of any integrable, compactly supported initial density towards a characteristically spreading self-similar profile. A potential theoretic comparison technique is outlined below which establishes the sharp <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo stretchy="false">/</mml:mo> <mml:mi>t</mml:mi> </mml:math> conjectured power law rate of decay uniformly in relative error, and in weaker norms such as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi mathvariant="bold">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> .
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.015 | 0.002 |
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 it