Gamba, Irene M.; Morawetz, Cathleen S. A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow: Existence theorem for potential flow. (English) Zbl 0863.76029 Commun. Pure Appl. Math. 49, No. 10, 999-1049 (1996). 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. Cited in 42 Documents MSC: 76H05 Transonic flows 35Q35 PDEs in connection with fluid mechanics 35Q60 PDEs in connection with optics and electromagnetic theory Keywords:boundary value problem; Poisson equation; velocity potential; electric potential × Cite Format Result Cite Review PDF Full Text: DOI