Radially symmetric spiral flows of the compressible Euler-Poisson system for semiconductors. (English) Zbl 1527.35419
Summary: In this paper, we study the steady flows to the compressible Euler-Poisson system for semiconductors with the nonzero angular velocity in a radially symmetric way in an annulus. The main purpose here is to elucidate the effect of the angular velocity in the structure of the steady flows. We show the well-posedness of all kinds of types of radially symmetric spiral flows including radial subsonic/supersonic/transonic flows, and further give a specific classification of the flow patterns under the assumption of various boundary conditions at the inner and the outer circle. Additionally, different from the purely radial case, the uniqueness of radial subsonic flow can not be obtained due to the nonlocal effect caused by the angular velocity, consequently we prove the uniqueness of the radial subsonic solution in the case without the semiconductor effect or with a small current assumption. Moreover, some new patterns of spiral flows with or without shock are observed, such as a smooth transonic flow and a supersonic-supersonic shock flow for a large relaxation time parameter.
MSC:
| 35Q81 | PDEs in connection with semiconductor devices |
| 35Q31 | Euler equations |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 76J20 | Supersonic flows |
| 76H05 | Transonic flows |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 78A35 | Motion of charged particles |
| 82D37 | Statistical mechanics of semiconductors |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 47H10 | Fixed-point theorems |
Keywords:
Euler-Poisson system; nonzero angular velocity; radially symmetric; nonlocal effect; smooth transonic flow; supersonic-supersonic shock flowReferences:
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