标题： 欧拉-泊松方程在声学边界条件下半导体亚音速稳态解的最优正则性 显示英文标题

标题： Optimal regularity of subsonic steady-states solution of Euler-Poisson equations for semiconductors with sonic boundary

摘要： 在本文中，我们研究了带有音速边界条件的半导体单极等温流体动力学模型的定常音速-亚音速解的最佳正则性。应用比较原理和能量估计，我们得到音速-亚音速解的正则性为$C^{\frac{1}{2}}[0,1]\cap W^{1,p}(0,1)$对于任何$p<2$，然后通过分析音速线上奇异点附近的解的性质，证明其为最优的，即$\rho\notin C^\nu[0,1]$对于任何$\nu>\frac{1}{2}$，以及$\rho\notin W^{1,\kappa}(0,1)$对于任何$\kappa\ge 2$。 此外，我们探讨半导体效应在声速点$x=1$和$x=0$处解的奇异性的影响，即对于任何松弛时间$\tau>0$，解在声速点$x=1$总是具有强奇异性，但是，一旦松弛时间足够大$\tau\gg 1$，那么声速-亚声速定态在两个声速边界$x=0$和$x=1$处都具有强奇异性。 我们还证明纯亚音速解 $\rho$属于 $W^{2,\infty}(0,1)$，它可以嵌入到 $C^{1,1}[0,1]$中，并且其正则性要优于音速-亚音速解的正则性。 显示英文摘要

摘要： 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., $\rho\notin C^\nu[0,1]$ for any $\nu>\frac{1}{2}$, and $\rho\notin W^{1,\kappa}(0,1)$ for any $\kappa\ge 2$. 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 $\tau>0$, but, once the relaxation time is sufficiently large $\tau\gg 1$, 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 $\rho$ 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.