An overview of some recent results on the Euler system of isentropic gas dynamics

In this overview we discuss some recent results of non--uniqueness for the isentropic Euler equations of gas dynamics with particular attention to the role of some admissibility criteria proposed in the literature.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1312.471

On the weak solutions to the equations of a compressible heat conducting gas

We consider the weak solutions to the Euler-Fourier system describing the motion of a compressible heat conducting gas. Employing the method of convex integration, we show that the problem admits infinitely many global-in-time weak solutions for any choice of smooth initial data. We also show that for any initial distribution of the density and temperature, there exists an initial velocity such that the associated initial-value problem possesses infinitely many solutions that conserve the total energy

On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System

In this paper we extend and complement the results in Chiodaroli etal. (Global ill-posedness of the isentropic system of gas dynamics, 2014) on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law p(ρ)=ρ γ , γ ≥1. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi (Ann Math 170:1417-1436, 2009), (Arch Ration Mech Anal 195:225-260, 2010). Moreover we prove that for some of these Riemann problems and for 1≤ γ <3 such solutions have a 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 (J Diff Equ 14:202-212, 1973) does not favour the classical self-similar solutions

Global ill-posedness of the isentropic system of gas dynamics

We consider the isentropic compressible Euler system in 2 space dimensions with pressure law $p({\rho}) = {\rho}^2$ and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions.Comment: 30 page

A-free rigidity and applications to the compressible Euler system

Can every measure-valued solution to the compressible Euler equations be approximated by a sequence of weak solutions? We prove that the answer is negative: generalizing a well-known rigidity result of Ball and James to a more general situation, we construct an explicit measure-valued solution for the compressible Euler equations which cannot be generated by a sequence of distributional solutions. We also give an abstract necessary condition for measure-valued solutions to be generated by weak solutions, relying on work of Fonseca and Müller. While a priori it is not unexpected that not every measure-valued solution arises from a sequence of weak solutions, it is noteworthy that this observation in the compressible case is in contrast to the incompressible situation, where every measure-valued solution can be approximated by weak solutions, as shown by Székelyhidi and Wiedemann. The final publication is available at Springer via http://dx.doi.org/10.1007/s10231-016-0629-9