## Abstract

The artificial compressible system gives a compressible approximation of the incompressible Navier–Stokes system. The latter system is obtained from the former one in the zero limit of the artificial Mach number ϵ which is a singular limit. The sets of stationary solutions of both systems coincide with each other. It is known that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion, then it is also stable as a solution of the artificial compressible one for sufficiently small ϵ. In general, the range of ϵ shrinks when the spectrum of the linearized operator for the incompressible system approaches to the imaginary axis. This can happen when a stationary bifurcation occurs. It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of ϵ can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.

Original language | English |
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Pages (from-to) | 1213-1228 |

Number of pages | 16 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 20 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sept 1 2018 |

Externally published | Yes |

## Keywords

- Artificial compressible system
- Bifurcation
- Incompressible Navier–Stokes system
- Singular perturbation
- Stability

## ASJC Scopus subject areas

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