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

Abstract

In this paper we extend and complement the results in Chiodaroli et al. (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(\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 infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi. Moreover we prove that for some of these Riemann problems and for 1\leq\gamma<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 does not favour the classical self-similar solutions

A counterexample to well-posedness of entropy solutions to the compressible Euler system

“…Very recently further investigations have been performed in order to understand the role of the initial data in the failure of uniqueness of entropy solutions and to explore other admissibility criteria: we refer the reader to [2,3,11] for more details.…”

Weak-strong uniqueness for measure-valued solutions of some compressible fluid models

“…Weak-strong uniqueness for compressible Euler models appears important in the light of several recent examples of non-uniqueness of admissible weak solutions [DLS10,Chi14,CK14,CFK15,CDLK14,Fei14]. For the Savage-Hutter equations, such examples were very recently constructed in [FGSG15].…”

High dimensionality and h-principle in PDE

“…In fact the same constructions can be used in compressible fluid dynamics to disprove the uniqueness of entropy admissible weak solutions for some regular (more precisely Lipschitz) initial data, cf. [19,20,35]. It is presently not known whether one could use techniques similar to those of [37] to construct continuous solutions which satisfy the local energy inequality (99).…”

“…In the case of Theorem 5.2(a) we can use a finite number of steps in the general Nash-Kuiper scheme to get a new short map from which we can proceed as in Section 4.3: taking advantage of the minimal decomposition (20) we reach the threshold…”

“…Since the number s n is the dimension of the space of symmetric matrices, it is clearly the minimal number for which the decomposition of Lemma 4.3 can be valid in C r 0 . A similar decomposition to (20), which is valid for all positive definite A, can also be proved using a locally finite partition of unity in the space of positive definite matrices (this is contained in Nash's paper [63], see also [32,78]), although then the sum in (20) is only locally finite and the number of nonvanishing terms is significantly larger than s n . Such a decomposition has also proved useful in other contexts; see [43,Lemma 17.13] and [60].…”