Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction (Q932844)

The asymptotic behavior of a spherically symmetric flow of a compressible viscous and polytropic ideal fluid in a field of external forces over an unbounded exterior domain of a sphere in \(n\)-dimensional space are considered. The unique existence of the stationary solution is shown under the adhesion and the adiabatic boundary conditions. Then, it is shown that a solution to the initial boundary value problem with the same boundary and spatial asymptotic conditions uniquely exists globally in time and converges to the stationary solution as time tends to infinity. In this stability theorem, the initial data can be chosen arbitrarily large as far as it belongs to suitable Sobolev space. Moreover, if the external force is attractive to the center of the sphere, it can be taken arbitrarily large. The proof of the stability theorem is based on the standard energy method using the Lagrangian coordinates. Here, it is the essential step to obtain the pointwise estimate of the density and the absolute temperature uniformly in time. The former is obtained through estimating the representation formula of density with the aid of the energy estimate.