Relaxation Time Limits of Subsonic Steady States for Hydrodynamic Model of Semiconductors with Sonic or Nonsonic Boundary
Bibliographic record
Abstract
Abstract. This paper concerns the relaxation time limits for the one-dimensional steady hydrodynamic model of semiconductors in the form of Euler–Poisson equations with sonic or nonsonic boundary. The sonic boundary is the critical and difficult case because of the degeneracy at the boundary and the formation of the boundary layers. In order to avoid the degeneracy of the second order derivatives, we technically introduce an invertible transform to the working equation. This guarantees that the remaining one order degeneracy becomes a good term since the transform used here is strictly increasing. Then we efficiently overcome the degenerate effect. When the relaxation time [Formula: see text], we first show the strong convergence of the approximate solutions to their asymptotic profiles in [Formula: see text] norm with the order [Formula: see text]. When [Formula: see text], the boundary layer appears because the boundary data are not equal to each other, and we further derive the uniform error estimates of the approximate solutions to their background profiles in [Formula: see text] norm with the order [Formula: see text] or [Formula: see text] according to the different cases of boundary data. Unlike the methods adopted in the previous studies, we propose some altogether new techniques of the asymptotic limit analysis to successfully describe the width of the boundary layer, which is almost the order [Formula: see text] provided [Formula: see text]. These original approaches develop and improve the existing studies. Finally, some numerical simulations are carried out, which confirm our theoretical study, in particular, the appearance of boundary layers.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".