Delta‐Shocks and Vacuums in Non‐Self‐Similar Riemann Solutions of the Umami Chaplygin Euler Equations With an Outer Force

This paper is concerned with the formation of delta‐shock and vacuum for the zero‐pressure Euler equations with friction by introducing the Umami Chaplygin gas model (a new variation of Chaplygin gas model) as the pressure perturbation. In order to match the Umami Chaplygin gas pressure, the reasonable flux‐function is introduced into the isentropic Euler system of gas dynamics with friction. Using variable substitution method, the Riemann problem for the perturbed model is solved with six kinds of structures. Intriguingly, the delta‐shock occurs in the solution in some cases, although the perturbed system is strictly hyperbolic with two genuinely nonlinear characteristic fields. For the delta‐shock, the generalized Rankine–Hugoniot relation and entropy condition are clarified. The solutions are no longer self‐similar due to the outer force. Moreover, it is illustrated that the non‐self‐similar solutions converge to those of the zero‐pressure Euler equations with friction as the Umami Chaplygin gas pressure and flux‐function approach to zero simultaneously.