The limits of Riemann solutions to the relativistic van der Waals fluid

Abstract

By a Riemann problem, we identify and analyze the phenomena of concentration and cavitation and the formation of delta-shock waves and vacuum states in solutions to the relativistic van der Waals fluid as the pressure and molecular excluded volume drop to zero simultaneously. Particularly, the geometric properties of elementary wave curves and the formation process of delta shocks and vacuums, influenced by the parameter of pressure perturbation ε and the van der Waals excluded volume b, are characterized and clarified completely. It is rigorously shown that the Riemann solution of the relativistic van der Waals fluid containing two shocks and two-rarefaction waves respectively tend to the delta-shock solution and vacuum state to the pressureless relativistic Euler equations as both pressure and molecular excluded volume vanish.