Existence of global entropy solutions to the isentropic Euler equations with geometric effects

In 2011, by Y.-G. Lu [Y.-G. Lu, Global existence of solutions to resonant system of isentropic gas dynamics, Nonlinear Anal. RWA 12 (2011) 2802–2810], the maximum principle was used to study the uniformly bounded L∞L∞ estimates z(ρδ,ε,uδ,ε)≤B(x),w(ρδ,ε,uδ,ε)≤M(t)z(ρδ,ε,uδ,ε)≤B(x),w(ρδ,ε,uδ,ε)≤M(t) for the εε-viscosity and δδ-flux-approximation solutions (ρδ,ε,uδ,ε)(ρδ,ε,uδ,ε) of the nonhomogeneous system (1.3), where w,zw,z are Riemann invariants of (1.3) and M(t)M(t) depends on the bound of the nonlinear function a(x)a(x), which excludes the class of discontinuous functions. In this short paper, we obtain the estimate w(ρδ,ε,uδ,ε)≤βw(ρδ,ε,uδ,ε)≤β when a′(x)≥0a′(x)≥0 for a suitable constant ββ depending only on the bound of a(x)a(x) and prove the existence of bounded entropy solutions, for the Cauchy problem of the isentropic Euler equations with geometric effects (1.1), which extend the results of the finite energy solution in LeFloch and Westdickenberg [P. LeFloch, M. Westdickenberg, Finite energy solutions to the isentropic Euler equations with geometric effects, J. Math. Pures Appl. 88 (2007) 389–429] and weak solutions in Tsuge [N. Tsuge, Global L∞L∞ Solutions of the Compressible Euler Equations with Spherical Symmetry, J. Math. Kyoto Univ. 46(2006) 457–524] for a polytropic gas with γ∈(1,53] to the general pressure function P(ρ)P(ρ).