##
Formation of Delta-Shocks and Vacuum States in the Vanishing Pressure
Limit of Solutions to the Isentropic Euler Equations

#### Authors: Gui-Qiang Chen and Hailiang Liu

#### Title:
Formation of Delta-Shocks and Vacuum States in the Vanishing Pressure
Limit of Solutions to the Isentropic Euler Equations

**Abstract**

The phenomena of concentration and cavitation and the formation of
$\delta$-shocks and vacuum states in the vanishing pressure limit are
identified and analyzed in inviscid compressible fluid flow.
It is shown that any two-shock Riemann solution of the Euler equations
for isentropic fluids tends to a $\delta$-shock solution of the Euler
equations for pressureless fluids, and the intermediate density between
the two shocks tends to a $\delta$-mass that forms the $\delta$-shock;
by contrast, any two-rarefaction-wave Riemann solution for isentropic
fluids tends to a two-contact-discontinuity solution whose
intermediate state between the two contact discontinuities is a vacuum
state for pressureless fluids,
even when the initial data stay away from the vacuum.
Some numerical results, which exhibit the formation processes of
$\delta$-shocks and vacuum states, are presented.

This article will be appeared in:

* SIAM J. Mathematical Analysis
* ** vol. 34**,
pages 925-938 (2003)

This paper is available in the following formats:

A closely related paper is Change me.

**Author Address**
Gui-Qiang Chen
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu