We prove the existence and uniqueness of the Riemann solutions to the Euler equations closed by N independent constitutive pressure laws. This model stands as a natural asymptotic system for the multi-pressure Navier–Stokes equations in the regime of infinite Reynolds number. Due to the inherent lack of conservation form in the viscous regularization, the limit system exhibits measure-valued source terms concentrated on shock discontinuities. These non-positive bounded measures, called kinetic relations, are known to provide a suitable tool to encode the small-scale sensitivity in the singular limit. Considering N independent polytropic pressure laws, we show that these kinetic relations can be derived by solving a simple algebraic problem which governs the endpoints of the underlying viscous shock profiles, for any given but prescribed ratio of viscosity coefficient in the viscous perturbation. The analysis based on traveling wave solutions allows us to introduce the asymptotic Euler system in the setting of piecewise Lipschitz continuous functions and to study the Riemann problem.