Smooth Transonic Steady-States Of Hydrodynamic Model For Semiconductors

In this paper,we investigate the existence and regularity of the smooth tran-sonic steady solutions of Euler-Poisson equations representing for hydrodynamic model of semiconductors.As the crucial mechanism to affect the structure of the stationary Euler-Poisson equations will be the doping profile,we divide it into two cases.In this paper,the doping profile is supposed a constant and then it is au-tonomous,so we do not consider the boundary effects.When the doping profile is supersonic,we prove that the Euler-Poisson system posses two C~?-smooth tran-sonic solutions.One is from supersonic region to 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 supersonic region 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 C~m-smooth.The method adopted is mainly the manifold analysis and singularity analysis near the sonic line and the singular point.