SMOOTH TRANSONIC STEADY STATES OF HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

In this paper, we investigate the existence and regularity of the smooth transonic steady solutions of Euler–Poisson equations representing for the hydrodynamic model of semiconductors. Different from the previous studies with the various setting boundaries, we observe that the crucial mechanism to affect the structure of the stationary Euler–Poisson equations is the doping profile. When the doping profile is supersonic, regardless of the boundary settings, we prove that the Euler–Poisson system possesses two C∞-smooth transonic solutions. One is from the supersonic region to the subsonic region, and the other is of the inverse direction. However, when the doping profile is subsonic, the case is more complicated. We prove that there is no continuous transonic solution if the semiconductor effect is small enough, but there will arise two kinds of smooth transonic solutions when the semiconductor effect is large enough. Both of them are from the supersonic region the to subsonic region, where one is a unique C∞-smooth transonic solution with a relatively large number as its derivative at the sonic point, and the other consists of a class of smooth transonic solutions with another relatively small number as the derivative at the sonic point. This class of solutions are proved mostly to be C∞ smooth, except for a special case in which we only prove the Cm smoothness. The method adopted is mainly the manifold analysis and the singularity analysis near the sonic line and the singular point.