Existence theory for the isentropic Euler equations
The existence of entropy solutions to Euler equations \(\partial_t \phi+\partial_x (\phi v)=0\), \(\partial_t (\phi v) +\partial_x (\phi v^2 +p(\phi))=0\) modeling isentropic compressible fluids is proved under the standard assumption \(p'(\phi)>0\) for \(\phi>0\), where \(\phi \geq 0\) denotes the density, \(v\) the velocity, and \(p(\phi) \geq 0\) the pressure of the fluid. The authors develop a new approach for constructing mathematical entropies for Euler equations which are singular near the vacuum; in particular, they identify the optimal assumption required on the singular behavior of pressure law in the vacuum in order to validate the two-term asymptotic expansion of entropy kernel. For more general pressure laws, a new multiple-term expansion is introduced based on Bessel functions with suitable exponents, and an optimal assumption is identified needed to validate the multiple-term expansion and to establish the existence theory. The obtained results cover, as a special example, the density-pressure law \(p(\phi)=k_1 \phi^{\gamma_1}+ k_2 \phi^{\gamma_2}\) where \(\gamma_1,\gamma_2 >1 \) and \(k_1,k_2>0\) are arbitrary constants.
scientific article; zbMATH DE number 1922373