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Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. (English) Zbl 1098.76603
Summary: Numerical simulations [2-D Riemann problem in gas dynamics and formation of spiral, in: Nonlinear Problems in Engineering and Science–Numerical and Analytical Approach (Beijing, 1991), Science Press, Beijing, 1992, pp. 167-179; Discrete Contin. Dyn. Syst. 1, 555-584 (1995); 6, 419-430 (2000)] 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, 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 the 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.
MSC:
 76N15 Gas dynamics, general 35B25 Singular perturbations (PDE) 35L67 Shocks and singularities 35Q35 PDEs in connection with fluid mechanics 76L05 Shock waves; blast waves (fluid mechanics) 76M99 Basic methods in fluid mechanics