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

Abstract

We consider a class of “wild” initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural $$L^1$$ L 1 -topology and show that its complement is rather large, specifically it is an open dense set.

Glimm’s method and density of wild data for the Euler system of gas dynamics *

“…In particular, the Euler system admits wild data-the initial conditions that give rise to infinitely many physically admissible solutions. Nonetheless, the fact that the energy equality (1.3) is included in the system may give some hope that wild data may be somehow 'exceptional' for the complete Euler system, see [11]. In this paper, we show that the wild data for the Euler system with periodic boundary conditions are dense in the L p −topology.…”

Localized mixing zone for Muskat bubbles and turned interfaces

“…[49,10,34]), the evolution of active scalars ( [30,68,52,50,11,47]) and transport equations ( [33,60,61,59]), the compressible Euler equations (see e.g. [22,21,55,42,1,54]), the Navier-Stokes equations (see e.g. [13,9,23,14]) and Magnetohydrodynamics ( [40,8,41]) (see also the surveys [37,38,12] and the references therein).…”

On bounded two-dimensional globally dissipative Euler flows

“…With the onset of convex integration in fluid dynamics it was possible to show that wild initial data is dense in the set of all divergence-free velocity fields, see [12,13,34,37] for weakly admissible solutions of different regularity classes with [12] by Daneri, Runa, Székelyhidi covering the C 1/3− case in three space dimensions. The question has also been addressed for weakly admissible solutions of the compressible Euler equations, see [8,20].…”