Concentration and cavitation in the vanishing pressure limit of solutions to the generalized Chaplygin Euler equations of compressible fluid flow

Abstract

Abstract The phenomena of concentration and cavitation and the formation of delta shock waves and vacuum states in vanishing pressure limits of solutions to the generalized Chaplygin Euler equations of compressible fluid flow are analyzed. It is proved that, as the pressure vanishes, any two-shock-wave Riemann solution of the generalized Chaplygin Euler equations of compressible fluid flow tends to a delta-shock solution to the transport equations, and the intermediate density between them tends to a weighted δ -measure that forms a delta shock wave; any two-rarefaction-wave solution is shown to tend to two contact discontinuities connecting the constant states and vacuum states, which form a vacuum solution of the transport equations. Moreover, some numerical simulations completely coinciding with the theoretical analysis are presented.