A Functional Integral Approach to Shock Wave Solutions of the Euler Equations with Spherical Symmetry (II)

Abstract

This is a continuation of [15]. As is well known, one dimensional conservation laws without source term have been extensively investigated after the foundamental paper of J. Glimm [3]. And those with integrable source terms were solved by Liu [6, 7], etc. For higher dimensional case with spherical symmetry, Makino [8] first proved a linear growth rate for solutions whenP(ρ)=σ2ργ, whereγ=1. But in order to get global existence forγ≠1 and the decay property, we need to find a uniform bound for the approximate solutions. In [15], we introduced a new norm and a functional integral approach to prove a uniform bound for a model problem of Euler equation inR3with spherical symmetry. In order to overcome the geometric effects of spherical symmetry which leads to a non-integrable source term, we considered an infinite reflection problem and solved it by considering the cancellations between reflections of different orders. In this paper, we consider a system which describes the isentropic and spherically symmetric motion of gas flow surrounding a solid star with radius 1 and massM. It is interesting to note that the wave curves for this problem are no longer continuous and there is an extra term in the wave interaction estimates. By introducing a new norm, we prove a similar result as [15]. In the Appendix, we present a local existence theorem forγ≠1 which was also obtained by Makino [9] by different method. And we extend the results to the cases with different boundary conditions.