Relaxation Time Limits of Subsonic Steady States for Multidimensional Hydrodynamic Model of Semiconductors
Bibliographic record
Abstract
Abstract. This paper is concerned with the relaxation-time limits to a multidimensional radial steady hydrodynamic model of semiconductors in the form of Euler–Poisson equations with sonic or nonsonic boundary as the relaxation time [Formula: see text] and [Formula: see text], respectively, where the sonic boundary is the critical and difficult case, because of the degeneracy at the boundary and the formation of boundary layers. For the case of [Formula: see text], after showing the boundedness of the density by using the divergence form, we prove the convergence of the solutions to their nontrivial asymptotic states with the convergence order [Formula: see text] in the [Formula: see text]-sense. In order to overcome the degeneracy caused by the critical sonic boundary, we introduce an inverse transform as a technical tool to remove the second-order degeneracy, and observe the advantage of a first-order degeneracy due to the monotonicity of this transformation. Moreover, when [Formula: see text] with different boundary values, where the boundary layers appear, we show the strong convergence order [Formula: see text] or [Formula: see text] for different boundary cases. In order to overcome the difficulty caused by the boundary layer, we propose a new technique in asymptotic limit analysis and identify the width of the boundary layers as [Formula: see text]. These new proposed methods develop and improve upon the existing studies. Finally, a series of numerical simulations are conducted, which corroborate our theoretical analysis, particularly regarding the formation 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) | 0.001 | 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 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".