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In this paper, we study the limits of Riemann solutions to the inhomogeneousEuler equations of one-dimensional compressible fluid flow as the adiabaticexponent $\gamma$ tends to one. Different from the homogeneous equations, theRiemann solutions of the inhomogeneous system are non self-similar. It isrigorously shown that, as $\gamma$ tends to one, any two-shock Riemann solutiontends to a delta shock solution of the pressureless Euler system with a sourceterm, and the intermediate density between the two shocks tends to a weighted$\delta$-mesaure which forms the delta shock; while any two-rarefaction-waveRiemann solution tends to a two-contact-discontinuity solution of thepressureless Euler system with a source term, whose intermediate state betweenthe two contact discontinuities is a vacuum state. Moreover, we also give somenumerical results to confirm the theoretical analysis.

The pressureless limits of Riemann solutions to the Euler equations of one-dimensional compressible fluid flow with a source term