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

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.