On steady state Euler-Poisson models for semiconductors.

*(English)*Zbl 0755.35138(Author’s summary.) This paper is concerned with an analysis of the Euler-Poisson model for unipolar semiconductor devices in the steady state isentropic case. In the two-dimensional case, we prove the existence of smooth solutions under a smallness assumption on the prescribed outflow velocity (small boundary current) and, additionally, under a smallness assumption on the gradient of the velocity relaxation time. The latter assumption allows a control of the vorticity of the flow and the former guarantees subsonic flow. The main ingredient of the proof is a regularization of the equation for the vorticity.

Also, in the irrotational two- and three- dimensional cases we show that the smallness assumption on the outflow velocity can be replaced by a smallness assumption on the (physical) parameter multiplying the drift- term in the velocity equation. Moreover, we show that solutions of the Euler-Poisson system converge to a solution of the shift-diffusion model as this parameter tends to zero.

Also, in the irrotational two- and three- dimensional cases we show that the smallness assumption on the outflow velocity can be replaced by a smallness assumption on the (physical) parameter multiplying the drift- term in the velocity equation. Moreover, we show that solutions of the Euler-Poisson system converge to a solution of the shift-diffusion model as this parameter tends to zero.

Reviewer: P.Hillion (Le Vesinet)

##### MSC:

35Q60 | PDEs in connection with optics and electromagnetic theory |

35Q35 | PDEs in connection with fluid mechanics |

35B65 | Smoothness and regularity of solutions to PDEs |

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\textit{P. A. Markowich}, Z. Angew. Math. Phys. 42, No. 3, 389--407 (1991; Zbl 0755.35138)

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##### References:

[1] | P. A. Markowich, C. Ringhofer and C. Schmeiser,Semiconductor Equations, Springer Verlag, Wien-New York 1990. |

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[4] | David Gilbarg and Neil S. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd Ed. Springer-Verlag, Berlin 1983. · Zbl 0562.35001 |

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