Structural stability of subsonic steady states to the hydrodynamic model for semiconductors with sonic boundary
Bibliographic record
Abstract
Abstract The hydrodynamic model for semiconductors with sonic boundary, represented by Euler–Poisson equations, possesses the various physical steady states including interior-subsonic/interior-supersonic/shock-transonic/C 1-smooth-transonic steady states. Since these physical steady states result in some serious singularities at the sonic boundary (their gradients are infinity), this makes that the structural stability for these physical solutions is more difficult and challenging, and has remained open as we know. In this paper, we investigate the structural stability of interior subsonic steady states. Namely, when the doping profiles are as small perturbations, the differences between the corresponding subsonic solutions are also small. To overcome the singularities at the sonic boundary, we propose a novel approach, which combines the weighted multiplier technique, local singularity analysis, monotonicity argument and squeezing skill. Both the result itself and the technique developed here will give us some truly enlightening insights into our follow-up study on the structural stability of the remaining types of solutions. A number of numerical approximations are also carried out, which intuitively confirm our theoretical results.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".