Multiscale analysis for convection dominated transport equations
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Bibliographic record
Abstract
In this paper, we perform a systematic multiscale analysis forconvection dominated transport equations with a weak diffusion and ahighly oscillatory velocity field. The paper primarily focuses onupscaling linear transport equations. But we also discuss brieflyhow to upscale two-phase miscible flows, in which casethe concentration equation is coupled to the pressure equationin a nonlinear fashion. For the problem we consider here,the local Peclet number is of order $O(\epsilon^{-m+1})$ with $m \in[2,\infty]$ being any integer, where $\epsilon$ characterizes thesmall scale in the heterogeneous media. Due to the presence of thenonlocal memory effect, upscaling a convection dominated transportequation is known to be very difficult. One of the key ideas inderiving a well-posed homogenized equation for the convectiondominated transport equation is to introduce a projection operatorwhich projects the fluctuation onto a suitable subspace. Thisprojection operator corresponds to averaging along the streamlinesof the flow. In the case of linear convection dominated transportequations, we prove the well-posedness of the homogenized equationsand establish rigorous error estimates for our multiscale expansion.
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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) | 0.000 | 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) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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