### Abstract

Time periodic problem for the compressible Navier-Stokes equation for barotropic flow on the whole space is studied. The existence of a time periodic solution is proved for sufficiently small time periodic external force with some symmetry when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-. T-map associated with the linearized problem around the motionless state with constant density in some weighted Sobolev space. The stability of the time periodic solution is also proved and the decay estimate of the perturbation is established.

Original language | English |
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Pages (from-to) | 399-444 |

Number of pages | 46 |

Journal | Journal of Differential Equations |

Volume | 258 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2015 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

**Existence and stability of time periodic solution to the compressible Navier-Stokes equation for time periodic external force with symmetry.** / Kagei, Yoshiyuki; Tsuda, Kazuyuki.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 258, no. 2, pp. 399-444. https://doi.org/10.1016/j.jde.2014.09.016

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TY - JOUR

T1 - Existence and stability of time periodic solution to the compressible Navier-Stokes equation for time periodic external force with symmetry

AU - Kagei, Yoshiyuki

AU - Tsuda, Kazuyuki

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Time periodic problem for the compressible Navier-Stokes equation for barotropic flow on the whole space is studied. The existence of a time periodic solution is proved for sufficiently small time periodic external force with some symmetry when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-. T-map associated with the linearized problem around the motionless state with constant density in some weighted Sobolev space. The stability of the time periodic solution is also proved and the decay estimate of the perturbation is established.

AB - Time periodic problem for the compressible Navier-Stokes equation for barotropic flow on the whole space is studied. The existence of a time periodic solution is proved for sufficiently small time periodic external force with some symmetry when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-. T-map associated with the linearized problem around the motionless state with constant density in some weighted Sobolev space. The stability of the time periodic solution is also proved and the decay estimate of the perturbation is established.

UR - http://www.scopus.com/inward/record.url?scp=84922821770&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84922821770&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2014.09.016

DO - 10.1016/j.jde.2014.09.016

M3 - Article

AN - SCOPUS:84922821770

VL - 258

SP - 399

EP - 444

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -