Existence of Dafermos-maximal DMV solutions

Establish whether for any finite-energy initial data there exists a dissipative measure–valued solution of the compressible Euler system that is maximal with respect to Dafermos’ local entropy-production order \prec_{\rm Daf}.

Background

Dafermos’ criterion compares solutions locally at the first time of divergence via the right derivative of total entropy; maximal elements under this order are expected to be weak solutions (a result conditional on their existence).

While regularity of \prec_{\rm Daf}-maximal DMV solutions has been proved (they are weak solutions if they exist), the existence of such maximal DMV solutions for arbitrary finite-energy data is unknown.

References

Open problem IV (Existence of maximal solutions in the sense of Dafermos): Does the Euler system admit a \prec_{\rm Daf} - maximal DMV solution for any finite energy initial data?

— The Euler system of gas dynamics  (2603.29619 - Feireisl, 31 Mar 2026) in Section 3.2, “Dafermos’ maximality criterion”