Celebratio Mathematica — Morawetz

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I. M. Gamba and C. S. Morawetz: “A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow: Existence theorem for potential flow,” Comm. Pure Appl. Math. 49 : 10 (1996), pp. 999–1049. MR 1404324 Zbl 0863.76029 article

In this paper 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 \( \phi \), and electric potential \( \Phi \) in the bounded domain \( \Omega \) that are \( C^{1,\alpha}(\overline{\Omega}) \), \( C^{2,\alpha}(\overline{\Omega}) \), and \( W^{2,\alpha}(\overline{\Omega}) \) functions, respectively, such that \( \rho \), \( \phi \), \( \Phi \), the speed \( |\nabla\phi| \), 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.

I. M. Gamba and C. S. Morawetz: “Viscous approximation to transonic gas dynamics: Flow past profiles and charged-particle systems,” pp. 81–102 in Modelling and computation for applications in mathematics, science, and engineering (Evanston, IL, 3–4 May 1996). Edited by J. W. Jerome. Numerical Mathematics and Scientific Computation. Oxford University Press (New York), 1998. MR 1677377 Zbl 0938.76047 incollection

A boundary value problem in a domain \( \Omega \) is considered for a system of equations of Fluid-Poisson type, i.e. 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 present an overview of the problem and, in addition, we show an improvement of the lower bound for the density that controls the rate of approach to cavitation density by a quantity of the order of the viscosity parameter to the power that corresponds to the inverse of the enthalpy function.

This is a necessary step in the existing programs in order to show existence of a solution for the transonic flow problem.

Irene M. Gamba	Related
Joseph Walter Jerome	Related