Relaxation Time Limits of Subsonic Steady States for Hydrodynamic Model of Semiconductors with Sonic or Nonsonic Boundary

SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3452-3477, June 2024.
Abstract. This paper concerns the relaxation time limits for the one-dimensional steady hydrodynamic model of semiconductors in the form of Euler–Poisson equations with sonic or nonsonic boundary. The sonic boundary is the critical and difficult case because of the degeneracy at the boundary and the formation of the boundary layers. In order to avoid the degeneracy of the second order derivatives, we technically introduce an invertible transform to the working equation. This guarantees that the remaining one order degeneracy becomes a good term since the transform used here is strictly increasing. Then we efficiently overcome the degenerate effect. When the relaxation time [math], we first show the strong convergence of the approximate solutions to their asymptotic profiles in [math] norm with the order [math]. When [math], the boundary layer appears because the boundary data are not equal to each other, and we further derive the uniform error estimates of the approximate solutions to their background profiles in [math] norm with the order [math] or [math] according to the different cases of boundary data. Unlike the methods adopted in the previous studies, we propose some altogether new techniques of the asymptotic limit analysis to successfully describe the width of the boundary layer, which is almost the order [math] provided [math]. These original approaches develop and improve the existing studies. Finally, some numerical simulations are carried out, which confirm our theoretical study, in particular, the appearance of boundary layers.