Delta-shocks in a logarithmic compressible flow under the combined isothermal-Chaplygin pressure perturbation

In the previous work of Zhang et al. [Limits of Riemann solutions to the compressible fluid flow with logarithmic pressure. Appl Anal 2025;104(6):1036–1062], a delta shock wave was unexpectedly discovered in the Riemann solution of the Euler equations of compressible flow with logarithmic pressure – a phenomenon that cannot occur in the corresponding isentropic Euler system under the same pressure. In this paper, we introduce a combined pressure model incorporating the isothermal and Chaplygin gases to perturb the logarithmic gas. Using the vanishing pressure limit method, we analyze in detail the formation of delta shock waves and rarefaction waves in the Riemann solutions for the compressible fluid flow with logarithmic pressure. We achieve this by letting the isothermal and Chaplygin equations of state tend to zero, while keeping the logarithmic gas equation of state fixed in the total pressure term. Finally, the numerical simulations consistent with the theoretical analysis are presented.