Concentration and Cavitation in the Vanishing Pressure Limit
of Solutions to the Euler Equations for Nonisentropic Fluids
Authors: Gui-Qiang Chen and Hailiang Liu
Title:
Concentration and Cavitation in the Vanishing Pressure Limit
of Solutions to the Euler Equations for Nonisentropic Fluids
Abstract
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.
This article has appeared in:
Physica D. vol. ??, pages ?? (2004)
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