### Abstract

The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of R ^{2}. It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura-Nishida energy method.

Original language | English |
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Pages (from-to) | 585-650 |

Number of pages | 66 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 205 |

Issue number | 2 |

DOIs | |

Publication status | Published - Aug 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

### Cite this

**Asymptotic Behavior of Solutions to the Compressible Navier-Stokes Equation Around a Parallel Flow.** / Kagei, Yoshiyuki.

Research output: Contribution to journal › Article

*Archive for Rational Mechanics and Analysis*, vol. 205, no. 2, pp. 585-650. https://doi.org/10.1007/s00205-012-0516-5

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TY - JOUR

T1 - Asymptotic Behavior of Solutions to the Compressible Navier-Stokes Equation Around a Parallel Flow

AU - Kagei, Yoshiyuki

PY - 2012/8/1

Y1 - 2012/8/1

N2 - The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of R 2. It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura-Nishida energy method.

AB - The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of R 2. It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura-Nishida energy method.

UR - http://www.scopus.com/inward/record.url?scp=84864361782&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864361782&partnerID=8YFLogxK

U2 - 10.1007/s00205-012-0516-5

DO - 10.1007/s00205-012-0516-5

M3 - Article

AN - SCOPUS:84864361782

VL - 205

SP - 585

EP - 650

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -