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)

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