Global Existence for General Systems of Isentropic Gas Dynamics via a Weighted Pressure Perturbation Approach

We establish the global existence of weak entropy solutions for 1D isentropic gas dynamics with general pressure laws ($γ> 1$). To address vacuum degeneracy, we introduce a novel structural regularization via a "Synchronized Dual Translation" strategy. This approach offers a distinct advantage over the flux-modification method of Lu (2007): while Lu's method induces a structural mismatch requiring restrictive constraints on $P'''$ to control regularization artifacts, our construction preserves structural isomorphism with the standard Euler equations. Consequently, the approximate entropies satisfy a homogeneous Generalized Euler-Poisson-Darboux equation. This allows us to prove strong convergence under natural asymptotic assumptions, effectively eliminating the need for the technical higher-order derivative constraints required in prior works.