Existence of global solutions to isentropic gas dynamics equations with a source term

In this paper we prove existence of isentropic gas dynamic equations with a source term (1.2). To this end we construct a sequence of regular hyperbolic systems (1.1) to approximate the inhomogeneous system of isentropic gas dynamics (1.2). First, for each fixed approximation parameter δ and very general condition on P(ρ), we establish the existence of entropy solutions for the Cauchy problem (1.1) with bounded initial date (1.4). Second, letting ε = o(δ), we obtain a complete proof of the Hloc−1 compactness of weak entropy pairs of system (1.2) in the form η(ρ, u) = ρH(ρ, u) given in Chen-LeFloch (2003). Finally, for the conditions of P(ρ) given in Chen-LeFloch (2003), applied to the results in Theorems 1 and 2, we obtain the global existence of entropy solutions for the Cauchy problem (1.2) with bounded initial date (1.4).