Maximal Dissipation and Well-posedness for the Compressible Euler System

Abstract

We discuss the problem of well-posedness of the compressible (barotropic) Euler system in the framework of weak solutions. The principle of maximal dissipation introduced by C.M. Dafermos is adapted and combined with the concept of admissible weak solutions. We use the method of convex integration in the spirit of the recent work of C.DeLellis and L.Székelyhidi to show various counterexamples to well-posedness. On the other hand, we conjecture that the principle of maximal dissipation should be retained as a possible criterion of uniqueness as it is violated by the oscillatory solutions obtained in the process of convex integration.

Non-uniqueness of admissible weak solutions to the Riemann problem for isentropic Euler equations

“…at EPFL in January 2017. During the workshop Ideal Fluids and Transport at IMPAN in Warsaw (February [13][14][15] 2017) the authors learned about similar results achieved by Ch. Klingenberg and S. Markfelder from Würzburg University.…”

“…where ρ ± , v ± are constants. The Riemann problem (1.1)-(1.3) has been the building block in the construction of non-unique entropy solutions for the isentropic Euler equations starting from Lipschitz initial data in [6]) and also in [8] for the investigation on the effectiveness of the entropy dissipation rate criterion, as proposed by Dafermos in [9], for the same system of equations (see also [14] for complemetary results on the Dafermos criterion). It is well known that the Riemann problem (1.1)-(1.3) admits self-similar solutions (ρ, v)(x, t) := (r, w)(x 2 /t) and that uniqueness holds in the class of admissible solutions if we require them to be self-similar and to have locally bounded variation.…”

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].…”

Relaxation of the Boussinesq system and applications to the Rayleigh–Taylor instability

“…In particular the induced solutions are admissible with respect to the initial energy. The usage of maximal energy dissipation is motivated by the entropy rate admissibility criterion for hyperbolic conservation laws [13] and has been investigated in [27,34] in the context of Euler subsolutions emanating from vortex-sheet initial data, as well as in [9,19] for compressible Euler systems. As in [27] we focus here on maximal initial dissipation, i.e.…”