Calculus of Variations and Geometric Measure Theory

E. Chiodaroli - O. Kreml

On the energy dissipation rate of solutions to the compressible isentropic Euler system

created by chiodaroli on 17 Dec 2013
modified on 20 Apr 2015


Published Paper

Inserted: 17 dec 2013
Last Updated: 20 apr 2015

Journal: Archive for Rational Mechanics and Analysis
Volume: 214
Number: 3
Pages: 1019-1049
Year: 2014
Doi: 10.1007/s00205-014-0771-8
Links: link


In this paper we extend and complement some recent results by Chiodaroli, De Lellis and Kreml on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law $p(\rho) = \rho^\gamma$, $\gamma \geq 1$. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also in finitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Sz\'ekelyhidi. Moreover we prove that for some of these Riemann problems and for $1\leq \gamma < 3$ such solutions have greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos in does not favour the classical self-similar solutions.