### Abstract

The main concern of the present paper is to analyze the behavior of a boundary layer, called a sheath, which appears over a material in contact with a plasma. The well-known Bohm criterion claims the velocity of positive ions should be faster than a certain constant for the formation of a sheath. The behavior of positive ions is governed by the Euler-Poisson equations. Mathematically, the sheath is understood as a monotone stationary solution, whose existence and asymptotic stability in one-dimensional space were proved in Suzuki's previous work. However the stability was proved under the assumption stronger than the Bohm criterion. In the present paper, we refine these results by proving the stability theorem exactly under the Bohm criterion in the spatial dimension up to three. We also deal with the degenerate case in which the Bohm criterion is marginally fulfilled.

Original language | English |
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Pages (from-to) | 761-790 |

Number of pages | 30 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 44 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 |

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### Keywords

- Asymptotic behavior
- Bohm criterion
- Convergence rate
- Sheath
- Stationary solution
- Weighted energy method

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*44*(2), 761-790. https://doi.org/10.1137/110835657