EXISTENCE THEORY FOR THE ISENTROPIC EULER EQUATIONS
Authors: Gui-Qiang Chen
Title:
EXISTENCE THEORY FOR THE ISENTROPIC EULER EQUATIONS
Abstract
We establish an existence theorem for entropy solutions to
the Euler equations modeling isentropic compressible fluids.
We develop a new approach for constructing mathematical entropies
for the Euler equations, which are singular near the vacuum.
In particular, we identify the {\it optimal assumption\/} required
on the singular behavior on the pressure law at the vacuum
in order to validate the {\it two-term\/} asymptotic expansion of
the entropy kernel proposed earlier by the authors. For more general
pressure laws, we introduce a new {\it multiple-term\/} expansion
based on the Bessel functions with suitable exponents,
and we also identify the optimal assumption to valid the
multiple-term expansion and to establish the existence theory.
Our results cover, as a special example, the density-pressure law
$p(\rho) = \kappa_1 \, \rho^{\gam_1} + \kappa_2 \, \rho^{\gam_2}$
where $\gam_1, \gamma_2 \in (1,3)$ and $\kappa_1, \kappa_2 >0$ are
arbitrary constants.
This article has appeared in:
Archive Rational Mechanics and Analysis
vol. 166, pages 81-98 (2003)
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