DiPerna’s conjecture on extremal points of the DMV solution set

Prove that every extreme point of the set of dissipative measure–valued (DMV) solutions \mathcal{U}[_0, v_0, S_0, \mathcal{E}_0] (i.e., DMV solutions emanating from initial data (_0, v_0, S_0, E_0)) is a weak solution of the compressible Euler system in the sense of the distributional equations with entropy inequality and conserved total energy.

Background

The DMV framework captures limits of consistent approximations, including turbulent effects via a parametrized Young measure and a concentration measure. The solution set \mathcal{U} is convex and compact in suitable topologies.

DiPerna conjectured that the extremal points of \mathcal{U} correspond to genuine weak solutions, linking measure-valued and weak formulations. Establishing this would clarify the structure of \mathcal{U} and underpin selection procedures via extreme points.

References

A celebrated conjecture formulated originally by DiPerna [DiP2] reads: The extremal points of the set \mathcal{U}[_0, v_0, S_0, \mathcal{E}_0] are weak solutions solutions of the Euler system.

— The Euler system of gas dynamics  (2603.29619 - Feireisl, 31 Mar 2026) in Section 2.3, “Dissipative vs. weak solutions”