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

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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Author Address
    Department of Mathematics
    Northwestern University
    Evanston, IL 60208-2730