Optimal regularity of subsonic steady-states solution of Euler-Poisson equations for semiconductors with sonic boundary
Bibliographic record
Abstract
In this paper, we study the optimal regularity of the stationary sonic-subsonic solution to the unipolar isothermal hydrodynamic model of semiconductors with sonic boundary. Applying the comparison principle and the energy estimate, we obtain the regularity of the sonic-subsonic solution as $C^{\frac{1}{2}}[0,1]\cap W^{1,p}(0,1)$ for any $p<2$, which is then proved to be optimal by analyzing the property of solution around the singular point on the sonic line, i.e., $ρ\notin C^ν[0,1]$ for any $ν>\frac{1}{2}$, and $ρ\notin W^{1,κ}(0,1)$ for any $κ\ge 2$. Furthermore, we explore the influence of the semiconductors effect on the singularity of solution at sonic points $x=1$ and $x=0$, that is, the solution always has strong singularity at sonic point $x=1$ for any relaxation time $τ>0$, but, once the relaxation time is sufficiently large $τ\gg 1$, then the sonic-subsonic steady-states possess the strong singularity at both sonic boundaries $x=0$ and $x=1$. We also show that the pure subsonic solution $ρ$ belongs to $W^{2,\infty}(0,1)$, which can be embedded into $C^{1,1}[0,1]$, and it is much better than the regularity of sonic-subsonic solutions.
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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 | 0.002 |
| 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 | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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 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".