Global Existence of Classical Solutions and Optimal Decay Rate for Compressible Flows via the Theory of Semigroups

Yoshihiro Shibata () and Yuko Enomoto
Additional contact information
Yoshihiro Shibata: Waseda University, Department of Mathematics and Research Institute for Science and Engineering
Yuko Enomoto: Shibaura Institute of Technology, Department of Mathematical Sciences

Chapter 39 in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 2018, pp 2085-2181 from Springer

Abstract: Abstract 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 H3 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 H2 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 H2 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.

Date: 2018
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-13344-7_52

Ordering information: This item can be ordered from
http://www.springer.com/9783319133447

DOI: 10.1007/978-3-319-13344-7_52

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().