Delta-shock solution for the nonhomogeneous Euler equations of compressible fluid flow with Born–Infeld equation of state

Abstract

The Born–Infeld type fluid, which obeys the pressure–density relation where the pressure is positive, is introduced into the nonhomogeneous Euler equations of compressible fluid flow. It is discovered for the first time that, for the positive pressure, the delta-shock with Dirac delta function in density develops in the solutions, even though the considered system is strictly hyperbolic with two genuinely nonlinear characteristic fields. First, the Riemann problem for the considered system is solvable with five kinds of structures by variable substitution method. For the delta-shock, the generalized Rankine–Hugoniot relation and entropy condition are clarified. Then it is discovered that as $A\to 0$, the solution consisting of two shocks converges to the delta-shock solution of zero-pressure Euler equations with friction; the delta-shock solution converges to that of zero-pressure Euler equations with friction; the solution containing two rarefaction waves converges to the vacuum solution of zero-pressure Euler equations with friction. Finally, the theoretical analysis is validated by the numerical results.