Convexity and equality of limits of consistent approximations and DMV solutions

Ascertain whether the set of weak limits of consistent approximations to the compressible Euler system is convex and whether this set coincides with the full set of dissipative measure–valued solutions \mathcal{U}[_0, v_0, S_0, \mathcal{E}_0] for given initial data.

Background

Consistent approximations (e.g., vanishing dissipation or numerical schemes) generate sequences whose limits define DMV solutions capturing turbulence via oscillations and concentrations.

The question is whether every DMV solution arises as a limit of consistent approximations and whether the collection of such limits is convex. For incompressible Euler an affirmative result is known (Székelyhidi–Wiedemann), but the compressible case remains unresolved.