##
Vacuum States and Global Stability of Rarefaction Waves for
Compressible Flow

#### Author: Gui-Qiang Chen

#### Title:
Vacuum States and Global Stability of Rarefaction Waves for
Compressible Flow

**Abstract**

The global stability of rarefaction waves in a broad class of entropy
solutions in $L^\infty$ containing vacuum states is proved for the
compressible Euler equations for both one-dimensional isentropic
and non-isentropic fluids. Rarefaction waves are the unique case that
produces the vacuum states late time in the Riemann solutions when the
Riemann initial data are away from the vacuum.
Such rarefaction waves are also shown to be global attractors of
entropy solutions in $L^\infty$ with the vacuum whose initial data are
an $L^\infty\cap L^1$ perturbation of those of the rarefaction waves.
Since the instability of solutions containing the vacuum states for the
compressible Navier-Stokes equations, some techniques are presented to
estimate a lower bound of the density for multidimensional viscous
non-isentropic fluids with spherical symmetry between a solid core and
a free boundary connected to a surrounding vacuum state.
Our analysis works for the solutions with arbitrarily large oscillation.
In particular, no assumption of small oscillation and $BV$ regularity of
entropy solutions is made for the compressible Euler equations.

This article has appeared in:

*Methods and Applications of Analysis, the Millennium Issue,
Dedicated to Cathleen Morawetz on the Occasioh of her 75th Birthday*,
**vol. 7 **, pages 337-362 (2000)

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A closely related paper is Change me.

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