On a one-dimensional đŒ-patch model with nonlocal drift and fractional dissipation
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Abstract
We consider a one-dimensional nonlocal nonlinear equation of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript t Baseline u equals left-parenthesis normal upper Lamda Superscript negative alpha Baseline u right-parenthesis partial-differential Subscript x Baseline u minus nu normal upper Lamda Superscript beta Baseline u"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> â </mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> â </mml:mo> <mml:mi> α </mml:mi> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi mathvariant="normal"> â </mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo> â </mml:mo> <mml:mi> Îœ </mml:mi> <mml:msup> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ÎČ </mml:mi> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial _t u = (\Lambda ^{-\alpha } u)\partial _x u - \nu \Lambda ^{\beta }u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda equals left-parenthesis minus partial-differential Subscript x x Baseline right-parenthesis Superscript one half"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo> â </mml:mo> <mml:msub> <mml:mi mathvariant="normal"> â </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\Lambda =(-\partial _{xx})^{\frac 12}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the fractional Laplacian and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi> Îœ </mml:mi> <mml:mo> â„ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\nu \ge 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the viscosity coefficient. We primarily consider the regime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than alpha greater-than 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi> α </mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0>\alpha >1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to beta less-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo> †</mml:mo> <mml:mi> ÎČ </mml:mi> <mml:mo> †</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0\le \beta \le 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -patch models. In the critical and subcritical range <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 minus alpha less-than-or-equal-to beta less-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> â </mml:mo> <mml:mi> α </mml:mi> <mml:mo> †</mml:mo> <mml:mi> ÎČ </mml:mi> <mml:mo> †</mml:mo> <mml:mn>2</mml:mn> <
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it