Asymptotic behavior for doubly degenerate parabolic equations
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Bibliographic record
Abstract
We use mass transportation inequalities to study the asymptotic behavior for a class of doubly degenerate parabolic equations of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" mode="display"> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mi>div</mml:mi> <mml:mfenced separators="" open="{" close="}"> <mml:mi>ρ</mml:mi> <mml:mi>∇</mml:mi> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>*</mml:mo> </mml:msup> <mml:mfenced separators="" open="[" close="]"> <mml:mi>∇</mml:mi> <mml:mfenced separators="" open="(" close=")"> <mml:mi>F</mml:mi> <mml:mo>'</mml:mo> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> </mml:mfenced> </mml:mfenced> </mml:mfenced> <mml:mspace width="4pt"/> <mml:mi>in</mml:mi> <mml:mspace width="3.30002pt"/> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>,</mml:mo> <mml:mi>and</mml:mi> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>in</mml:mi> <mml:mspace width="3.30002pt"/> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , or a bounded domain of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> in which case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mi>∇</mml:mi> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>*</mml:mo> </mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>∇</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>'</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mspace width="0.277778em"/> <mml:mi>·</mml:mi> <mml:mspace width="0.277778em"/> <mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> . We investigate the case where the potential V is uniformly c -convex , and the degenerate case where V =0. In both cases, we establish an exponential decay in relative entropy and in the c -Wasserstein distance of solutions – or self-similar solutions – of (1) to equilibrium, and we give the explicit rates of convergence. In particular, we generalize to all p >1, the HWI inequalities obtained by Otto and Villani (J. Funct. Anal. 173 (2) (2000) 361–400) when p =2. This class of PDEs includes the Fokker–Planck, the porous medium, fast diffusion and the parabolic p -Laplacian equations.
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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 | 0.001 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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 it