### 抜粋

In this chapter, we provide a review of results on the global well-posedness and optimal decay rate of strong solutions to the compressible Navier-Stokes equations in several type of domains: (1) whole space (Theorems 6, 7, 8, 9, 10, 11, and 12), (2) exterior domains (Theorems 13 and 14), (3) half-space (Theorem 15), (4) bounded domains (Theorem 16), and (5) infinite layers. Global well-posedness for the compressible viscous barotropic fluid motion with nonslip boundary condition was for the first time proved in the early 1980s by Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) under the assumption that the H^{3} norm of the initial data is small. In Theorems 1, 2, 3, and 4, we revisit the same problem as in Matsumura and Nishida (Commun Math Phys 89:445- 464, 1983) under the weaker assumptions, namely, that the H^{2} norm of initial data is small. This is an improvement of the result in Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) in view of the regularity assumption of the initial data. To show the methods, we perform the proof of Theorems 1, 2, 3, and 4 in all essential details. In this process, the L_{p}-L_{q} decay properties of solutions to the linearized equations are proved by using the cutoff technique and combining the local energy decay and the result in the whole space. This result was first proved by Kobayashi and Shibata (Commun Math Phys 200:621-659, 1999) under some additional assumption, and in this chapter, this assumption is eliminated by using a bootstrap argument. In the final section of this chapter, the optimal decay rate of the H^{2} norm of solution of the nonlinear problem is proved by combining the L_{p}-L_{q}decay properties of the linearized equations with some energy inequality of exponential decay type under the assumption that the initial data belong to the intersection space of H2 and L1. The main idea of this part of the proof is to combine the L_{p}-L_{q} decay properties of the Stokes semigroup and some Lyapunov-type energy inequality.

元の言語 | English |
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ホスト出版物のタイトル | Handbook of Mathematical Analysis in Mechanics of Viscous Fluids |

出版者 | Springer International Publishing |

ページ | 2085-2181 |

ページ数 | 97 |

ISBN（電子版） | 9783319133447 |

ISBN（印刷物） | 9783319133430 |

DOI | |

出版物ステータス | Published - 2018 4 19 |

### ASJC Scopus subject areas

- Mathematics(all)
- Physics and Astronomy(all)
- Engineering(all)

## フィンガープリント Global existence of classical solutions and optimal decay rate for compressible flows via the theory of semigroups' の研究トピックを掘り下げます。これらはともに一意のフィンガープリントを構成します。

## これを引用

*Handbook of Mathematical Analysis in Mechanics of Viscous Fluids*(pp. 2085-2181). Springer International Publishing. https://doi.org/10.1007/978-3-319-13344-7_52