Unboundedness of Some Solutions to Isentropic Model Equations for the One Dimensional Periodic Motions of a Compressible Self-Gravitating Viscous Fluid

Abstract

We consider the initial value problem for the one dimensional system of equations of a compressible viscous fluid driven by a bounded periodic self-gravitation of the fluid. For the initial-boundary value problem on a finite interval with a general bounded forcing term Matsumura and Nishida (Periodic solutions of a viscous gas equation. In: Mimura M, Nishida T (eds) Recent Topics in nonlinear PDE IV. Lecture notes in numerical and applied analysis, vol 10. North-Holland, Amsterdam, pp 49–82, 1989) showed that the isothermal system has a unique global-in-time solution for any initial data, being bounded as well. For the isentropic system Matsumura and Yanagi (Commun Math Phys 175:259–274, 1996) observed that given any initial data and any bounded forcing term, if the adiabatic constant is chosen suitably close to one, then the boundedness of the solution remains true. Without such a choice of the adiabatic constant, however, it is unknown whether the solution is bounded or not. In this paper we focus on the structure of stationary solutions to the isentropic self-gravitational system and prove that a certain bounded stationary solution is lost when the average of the specific volume reaches a critical value. We then show that there dose exist an unbounded solution for the initial value problem when the average exceeds the critical value. We also present a sufficient condition for the unboundedness in terms of the initial values of an energy form.