We study the large-time behavior of a spherically symmetric motion of isentropic and compressible viscous gas in a field of potential force over an unbounded exterior domain in ℝn(n≥2). First, we show the unique existence of a stationary solution satisfying an adhesion boundary condition and a positive spatial asymptotic condition. Then, it is shown that the stationary solution becomes 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 to the center of a sphere, it can also be taken arbitrarily large. The proof of the stability theorem is based on computations, executed by using the Lagrangian coordinate. In the proof, it is the essential step to obtain the pointwise estimate for the density. It is derived through employing a representation formula of the density with the aid of the standard energy method. The Hölder regularity of the initial data is also required for translating the results in the Lagrangian coordinate to those in the Eulerian coordinate.
Exact Large Time Behavior of Spherically Symmetric Plasmas
