Structural stability of subsonic solutions to a steady hydrodynamic model for semiconductors: From the perspective of boundary data

Abstract

In this paper, we are concerned with a one-dimensional isothermal steady hydrodynamic model for semiconductors driven by boundary data. In the purely subsonic setting, we obtain the existence, uniqueness and structural stability of purely subsonic solutions. Moreover, when the boundary data range from the subsonic region to the sonic line, we further study the degenerate problem from the perspective of boundary data, and prove that there exists a unique interior subsonic solution to the degenerate problem. As a byproduct, we also establish the structural stability between purely subsonic solution and interior subsonic solution in a relatively weak sense. These results provide us with a completely new perspective to understand the singularity caused by the boundary degeneracy. A number of numerical simulations are also carried out, which confirm our theoretical results.