Nonuniqueness of Admissible Weak Solution to the Riemann Problem for the Full Euler System in Two Dimensions

Abstract

The question of well-and ill-posedness of entropy admissible solutions to the multidimensional systems of conservation laws has been studied recently in the case of isentropic Euler equations. In this context special initial data were considered, namely the 1D Riemann problem which is extended trivially to a second space dimension. It was shown that there exist infinitely many bounded entropy admissible weak solutions to such a 2D Riemann problem for isentropic Euler equations, if the initial data give rise to a 1D self-similar solution containing a shock. In this work we study such a 2D Riemann problem for the full Euler system in two space dimensions and prove the existence of infinitely many bounded entropy admissible weak solutions in the case that the Riemann initial data give rise to the 1D self-similar solution consisting of two shocks and possibly a contact discontinuity.

On the density of “wild” initial data for the compressible Euler system

“…Recently, however, the method of convex integration developed in the context of the incompressible Euler system by De Lellis and Székelyhidi [7] has been adapted to identify a class of initial data for which the problem (1-3) admits infinitely many admissible weak solutions defined on a given time interval (0, T ), [9]. Similar results have been obtained also for the associated Riemann problem in [1].…”

“…The second convex integration ansatz, that is available in the literature, is based on the analysis of the corresponding Riemann problem, see [1,6,10] among others. We will first focus on the ansatz (10) and afterwards extend our results to the wild solutions obtained via the Riemann problem in Sect.…”

“…The first method [9] was explained above, see (10). The second method considers initial data constant in each of the two half spaces (Riemann data), see [1] and also [5,6,10] for the isentropic case. Here-instead of (10)-Eqs.…”

“…If the functions , u, and ϑ are continuously differentiable, the relations (1) give rise to the entropy balance:…”

A General Convex Integration Scheme for the Isentropic Compressible Euler Equations

“…In fact, the set of initial data that admits such nonunique solutions is dense in the energy space [5]. Similar constructions also work for related systems, such as the full compressible Euler system in multiple space dimensions [2,8,16]. All these constructions rely on a technique known as convex integration, introduced to the context of fluid dynamics in the seminal work of De Lellis-Székelyhidi [11,12].…”