A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow: Existence theorem for potential flow

In this paper we solve a boundary value problem in a two-dimensional domain O for a system of equations of Fluid-Poisson type, that is, a viscous approximation to a potential equation for the velocity coupled with an ordinary differential equation along the streamlines for the density and a Poisson equation for the electric field. A particular case of this system is a viscous approximation of transonic flow models. The general case is a model for semiconductors. We show existence of a density ρ, velocity potential φ, and electric potential Φ in the bounded domain O that are C^1,α(O¯), C^2,α(O¯), and W^2,α(O¯) functions, respectively, such that ρ, φ, Φ, the speed |Δφ|, and the electric field E = ΔΦ are uniformly bounded in the viscous parameter. This is a necessary step in the existing programs in order to show existence of a solution for the transonic flow problem. © 1996 John Wiley & Sons, Inc.