Global existence of classical solutions and optimal decay rate for compressible flows via the theory of semigroups

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_qdecay 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.