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A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow: Existence theorem for potential flow. (English) Zbl 0863.76029

Summary: We solve a boundary value problem in a two-dimensional domain \(\Omega\) 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 \(\rho\), velocity potential \(\varphi\), and electric potential \(\Phi\) in the bounded domain \(\Omega\) that are \(C^{1,\alpha} (\overline{\Omega})\), \(C^{2,\alpha} (\overline{\Omega})\), and \(W^{2,p} (\overline{\Omega})\) functions, respectively, such that \(\rho,\varphi,\Phi\), the speed \(|\nabla \varphi|\), and the electric field \(E=\nabla\Phi\) 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.

MSC:

76H05 Transonic flows
35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
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