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A steady state potential flow model for semiconductors. (English) Zbl 0808.35150
The paper concerns the hydrodynamic equations modelling a unipolar semiconductor. Here the unknown functions are $$\rho$$ = electron density, $$u=\{ u_ 1, u_ 2, u_ 3\}$$ = electron velocity and $$\varphi$$ = electrostatic potential. Under the assumption of potential flow $$u= - \nabla\psi$$ ($$\psi$$ = velocity potential) the equations take the form ${\textstyle {1\over 2}}| \nabla\psi |^ 2+ h(\rho)= \varphi+ {\textstyle {\psi\over\tau}}, \qquad \text{div}(\rho \nabla\psi)=0, \quad \Delta\varphi= \rho-C$ which have to be considered in a bounded domain $$\Omega\subset \mathbb{R}^ 3$$, where $$h$$ = given function, $$\tau= \text{const}>0$$ relaxation time and $$C= C(x)$$ $$(x\in\Omega)$$ given doping profile.
By the aid of Schauder’s fixed point theorem the author prove the existence of a solution $$(\psi,\rho, \varphi)\in C^{2,\delta} (\overline{\Omega})\times W^{2,q}(\Omega)\times W^{2,q} (\Omega)$$ $$(1\leq q<\infty)$$ provided $$\Omega$$ is a bounded convex $$C^{2,\delta}$$-domain in $$\mathbb{R}^ 3$$, $$\psi$$, $$\rho$$, $$\varphi$$ are subject to Dirichlet conditions along the whole boundary $$\partial\Omega$$ and an appropriate smallness condition on the Dirichlet datum of $$\psi$$ is satisfied. This smallness condition implies ellipticity of the problem under consideration which is equivalent to the case of subsonic flow of electrons in the semiconductor device.
Reviewer: J.Naumann (Berlin)

##### MSC:
 35Q60 PDEs in connection with optics and electromagnetic theory 78A35 Motion of charged particles 76G25 General aerodynamics and subsonic flows 35Q35 PDEs in connection with fluid mechanics
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##### References:
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