DiPerna’s conjecture on maximal entropy production

Show that any DMV solution in the set of dissipative measure–valued solutions \mathcal{U}[_0, v_0, S_0, \mathcal{E}_0] that is maximal with respect to DiPerna’s entropy-based order relation \prec_{\rm DiP} is a weak solution of the compressible Euler system in the sense of the distributional equations with entropy inequality and conserved total energy.

Background

DiPerna proposed a global-in-time entropy maximization order \prec_{\rm DiP}, comparing total entropies at each time. The conjecture asserts that DMV solutions maximal under this order are actually weak solutions.

Although \prec_{\rm DiP}-maximal DMV solutions exist, it is unresolved whether maximality enforces the collapse of measure-valued structure (no oscillations/concentrations) into a weak solution; only asymptotic regularity results are known.

References

Open problem III (DiPerna' s conjecture on maximal entropy production): Any DMV solution in \mathcal{U}[0, v_0, S_0, \mathcal{E}_0] that is maximal with respect to the order relation \prec{\rm DiP} is a weak solution of the Euler system in the sense of d29--d32. Although the existence of \prec_{\rm DiP}-maximal solutions was shown in , DiPerna's conjecture remains open.

— The Euler system of gas dynamics  (2603.29619 - Feireisl, 31 Mar 2026) in Section 3.1, “DiPerna’s maximality criterion”