### Abstract

Stability of stationary solutions of the Oberbeck–Boussinesq system (OB) and the corresponding artificial compressible system is considered. The latter system is obtained by adding the time derivative of the pressure with small parameter ε> 0 to the continuity equation of (OB), which was proposed by A. Chorin to find stationary solutions of (OB) numerically. Both systems have the same sets of stationary solutions and the system (OB) is obtained from the artificial compressible one as the limit ε→ 0 which is a singular limit. It is proved that if a stationary solution of the artificial compressible system is stable for sufficiently small ε> 0 , then it is also stable as a solution of (OB). The converse is proved provided that the velocity field of the stationary solution satisfies some smallness condition.

Original language | English |
---|---|

Pages (from-to) | 345-365 |

Number of pages | 21 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 19 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Mathematical Physics
- Condensed Matter Physics
- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Mathematical Fluid Mechanics*,

*19*(2), 345-365. https://doi.org/10.1007/s00021-016-0284-3

**On Chorin’s Method for Stationary Solutions of the Oberbeck–Boussinesq Equation.** / Kagei, Yoshiyuki; Nishida, Takaaki.

Research output: Contribution to journal › Article

*Journal of Mathematical Fluid Mechanics*, vol. 19, no. 2, pp. 345-365. https://doi.org/10.1007/s00021-016-0284-3

}

TY - JOUR

T1 - On Chorin’s Method for Stationary Solutions of the Oberbeck–Boussinesq Equation

AU - Kagei, Yoshiyuki

AU - Nishida, Takaaki

PY - 2017/6/1

Y1 - 2017/6/1

N2 - Stability of stationary solutions of the Oberbeck–Boussinesq system (OB) and the corresponding artificial compressible system is considered. The latter system is obtained by adding the time derivative of the pressure with small parameter ε> 0 to the continuity equation of (OB), which was proposed by A. Chorin to find stationary solutions of (OB) numerically. Both systems have the same sets of stationary solutions and the system (OB) is obtained from the artificial compressible one as the limit ε→ 0 which is a singular limit. It is proved that if a stationary solution of the artificial compressible system is stable for sufficiently small ε> 0 , then it is also stable as a solution of (OB). The converse is proved provided that the velocity field of the stationary solution satisfies some smallness condition.

AB - Stability of stationary solutions of the Oberbeck–Boussinesq system (OB) and the corresponding artificial compressible system is considered. The latter system is obtained by adding the time derivative of the pressure with small parameter ε> 0 to the continuity equation of (OB), which was proposed by A. Chorin to find stationary solutions of (OB) numerically. Both systems have the same sets of stationary solutions and the system (OB) is obtained from the artificial compressible one as the limit ε→ 0 which is a singular limit. It is proved that if a stationary solution of the artificial compressible system is stable for sufficiently small ε> 0 , then it is also stable as a solution of (OB). The converse is proved provided that the velocity field of the stationary solution satisfies some smallness condition.

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U2 - 10.1007/s00021-016-0284-3

DO - 10.1007/s00021-016-0284-3

M3 - Article

AN - SCOPUS:85019268141

VL - 19

SP - 345

EP - 365

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

SN - 1422-6928

IS - 2

ER -