Asymptotic behavior of spherically symmetric solutions to the compressible Navier Stokes equation with external forces
Nishibata, Shinya Asymptotic behavior of spherically symmetric solutions to the compressible Navier Stokes equation with external forces. In: The 28th Sapporo Symposium on Partial Differential Equations.
24 July 2003: The 28th Sapporo Symposium on Partial Differential Equations
We study the large time behavior of an isentropic and spherically symmetric motion of compressible viscous gas in a eld of external force over an unbounded exterior domain in R^n (n \ge 2). The typical example of this problem appears in analysis of the behavior of atmosphere around the earth. First, we show that there exists a stationary solution satisfying an adhesion boundary condition and a positive spatial asymptotic condition. Then, it is shown that this stationary solution is a time asymptotic state to the initial boundary value problem with the same boundary and spatial asymptotic conditions. Here, the initial data can be chosen arbitrarily large if it belongs to the suitable Sobolev space. Moreover, if the external force is attractive, it also can be arbitrarily large. This condition includes the most typical external force, i.e., the gravitational force. In the proof of the stability theorem, it is the essential step to obtain the uniform positive lower bound for the density. It is derived through the energy method with the aid of a representation formula for the density.