The Riemann Problem for a 3x3 Generalized Chaplygin Gas System with Variable Pressure

Abstract

We consider the Riemann problem for a 3x3 system of conservation laws with generalized Chaplygin pressure $p(ρ,v)=-\frac{A(v)}{ρ^α}$, $0<α\leq 1$, where the pressure depends on an additional transported variable. We analyze the system&#39;s wave structure and classify the Riemann solutions. Whenever classical solutions consisting of shocks, rarefaction waves, and contact discontinuities fail to exist, singular solutions arise. We verify that these satisfy the conservation laws in the distributional sense within the classical Dirac delta framework, and compare them with Nedeljkov&#39;s shadow-wave construction, giving two complementary descriptions of the same singular solution. We further study admissibility via the Dafermos maximum entropy dissipation principle, with several examples showing how it selects the physically relevant solution. Lax-Friedrichs simulations illustrate the Riemann wave patterns and provide a comparison with the analytical results. To construct viscous profiles for the isolated overcompressive $δ$-shock, we assume $α\in\mathbb{Q}\cap(0,1]$ and apply the Dafermos regularization together with a spherical blow-up. Working in three directional charts, we construct the reduced singular concatenation consisting of the left outer orbit, the middle orbit on the blown-up boundary, and the right outer orbit. We then prove that, for sufficiently small positive viscosity, this singular concatenation perturbs to a heteroclinic orbit. Consequently, the isolated overcompressive $δ$-shock is realized as the zero-viscosity limit of a family of self-similar Dafermos viscous profiles.