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:

This paper is available in the following formats:

A closely related paper is change me.

Author AddressGui-Qiang Chen Department of Mathematics Northwestern University Evanston, IL 60208-2730 USA gqchen@math.northwestern.edu