Global existence of solutions to resonant system of isentropic gas dynamics

In this paper, we study the global existence of weak solutions for the Cauchy problem of nonhomogeneous system of isentropic gas dynamics (1.1) with bounded initial data (1.2). First, we use the maximum principle to obtain the uniformly bounded L∞L∞ estimate z(ρδ,ε,uδ,ε)≤(B(x),wρδ,ε,uδ,ε)≤M(t)z(ρδ,ε,uδ,ε)≤(B(x),wρδ,ε,uδ,ε)≤M(t) for the εε-viscosity and δδ-flux-approximation solutions of (1.1) without the restriction z0(x)≥0z0(x)≥0 or w0(x)≤0w0(x)≤0 as given in the paper Klingenberg and Lu (1997) [8], where w,zw,z are Riemann invariants of (1.1) and B(x)>0B(x)>0 is a bounded function of xx depending on the function a(x)a(x) given in (1.1), but independent of δ,εδ,ε. Second, we use the compensated compactness theory and the compact frameworks given in Chen and LeFloch (2003) [18], Ding et al. (1989) [19], DiPerna (1983) [15], Lions et al. (1996) [16], Lions et al. (1994) [17] to prove the global existence of weak solutions for the Cauchy problem (1.1)–(1.2). Third, we extend the existence results of weak solutions for a polytropic gas with γ∈(1,53] in Tsuge (2006) [10] to the general pressure function P(ρ)P(ρ).