Global classical solutions with large initial data in the full compressible MHD system

Determine whether the full non-isentropic compressible magnetohydrodynamic system admits globally existing and unique classical solutions for large initial data, even in one spatial dimension when the density is bounded away from vacuum.

Background

The paper surveys unresolved issues for the full, non-isentropic compressible MHD equations when hydrodynamic and magnetic effects interact. Although global weak solutions and local strong solutions are cited for broad classes of data, the corresponding global classical theory for large data is described as unresolved.

The question remains open even in one spatial dimension under the comparatively favorable assumption that the density stays bounded away from vacuum, highlighting that large-data global regularity is substantially more difficult than the small-perturbation result proved in the paper.

References

Here the interaction between the hydrodynamic and the electrodynamic effects generates additional obstructions, and many basic questions remain open --- for instance, the global existence and uniqueness of classical solutions with large initial data is unresolved even in one space dimension for densities bounded away from vacuum.

— Stabilization by a background magnetic field: global well-posedness of the full compressible viscous non-resistive MHD system without heat-conductivity  (2609.01488 - Qiao et al., 1 Sep 2026) in Section 1, subsection “Known results,” paragraph “The full (non-isentropic) case”