Global existence of entropy-admissible weak solutions

Determine whether the full compressible Euler system of gas dynamics admits a globally defined entropy-admissible weak solution for any finite-energy initial data.

Background

Classical solutions of the Euler system can form shocks in finite time, necessitating weak solutions. However, convex integration results show severe non-uniqueness, even among entropy-admissible weak solutions, and the existence of such solutions for general data is unclear. The authors therefore pose a basic existence question for entropy-admissible weak solutions given arbitrary finite-energy initial data.

This problem probes whether the entropy inequality (Second Law) plus finite energy suffices to guarantee global-in-time weak solutions, despite known ill-posedness phenomena and wild initial data.

References

Despite the large number of recent results concerning the existence of wild data and infinitely many weak solutions to the Euler system, the existence of an entropy admissible solution for arbitrary, say bounded, initial data remains open. Specifically, does the Euler system admit a globally defined entropy admissible weak solution for any finite energy initial data?

— The Euler system of gas dynamics  (2603.29619 - Feireisl, 31 Mar 2026) in Subsection “Well posedness for the Euler system?”, Section 1 (Introduction)