Numerical simulations in \cite{CCY91,CCY95} for the Euler equations for gas dynamics in the regime of small pressure showed that, for one case, the particles seem to be more sticky and tend to concentrate near some shock locations; % which move with the associated shock speeds; and for the other case, in the region of rarefaction waves, the particles seem to be far apart and tend to form cavitation in the region. In this paper we identify and analyze the phenomena of concentration and cavitation by studying vanishing pressure limit of solutions of the full Euler equations for nonisentropic compressible fluids with a scaled pressure. It is rigorously shown that any Riemann solution containing two shocks and possibly one contact discontinuity to the Euler equations for nonisentropic fluids tends to a $\delta$-shock solution to the corresponding transport equations, and the intermediate densities between the two shocks tend to a weighted $\delta$-measure that, along with the two shocks and possibly contact discontinuity, forms the $\delta$-shock as the pressure vanishes. By contrast, it is also shown that any Riemann solution containing two rarefaction waves and possibly one contact discontinuity to the Euler equations for nonisentropic fluids tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate states between the two rarefaction waves tend to a vacuum state as the pressure vanishes. Some numerical results exhibiting the processes of concentration and cavitation are presented as the pressure decreases.

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Author AddressGui-Qiang Chen Department of Mathematics Northwestern University Evanston, IL 60208-2730 USA gqchen@math.northwestern.edu