Global-in-time strong solutions for compressible Navier–Stokes equations with large initial data

Establish global-in-time existence of strong solutions to the compressible Navier–Stokes equations for large initial data.

Background

The paper situates its contribution within the existing theory for compressible Navier–Stokes equations. Although local-in-time existence and uniqueness of strong solutions for large initial data are known, the corresponding global-in-time result remains unresolved. This problem concerns strong, rather than finite-energy weak, solutions and is presented as a known open problem in the background literature.

The paper itself addresses global-in-time existence of finite-energy weak solutions for a relaxed Navier–Stokes–Korteweg system, not the unresolved global strong-solution problem for the compressible Navier–Stokes equations. Accordingly, the problem is included as an explicitly stated open problem from the paper’s contextual discussion.

References

For large initial data, the local-in-time existence and uniqueness of strong solutions was proved in , while the corresponding global-in-time existence is an open problem.

— Global-in-time existence of finite energy weak solutions to a relaxed Navier-Stokes-Korteweg model  (2608.12969 - Oschmann et al., 13 Aug 2026) in Section 1, Introduction, paragraph beginning “Without being exhaustive, we give a brief overview on existence results for the rNSKE and related models.”

However, the regularities and uniquness of such global weak solutions generally remain open in multi-dimensional domains.

— Global well-posedness of radially symmetric strong solutions to two-dimensional compressible liquid crystal flows with large data and vacuum  (2609.00627 - Mei et al., 1 Sep 2026) in Section 1, Introduction and main result

However, whether this radial symmetric assumption can be removed in the case β>1 is still a challenge open problem even for a two-dimensional periodic domain.

— Global well-posedness of radially symmetric strong solutions to two-dimensional compressible liquid crystal flows with large data and vacuum  (2609.00627 - Mei et al., 1 Sep 2026) in Section 1, Introduction and main result