Global small solutions for the compressible non-resistive system on \(\mathbb{R}^3\)

Determine whether the compressible non-resistive magnetohydrodynamic system admits global small solutions on \(\mathbb{R}^3\) without any additional structural assumption.

Background

The paper discusses the loss of global well-posedness and stability theory when dissipative mechanisms such as magnetic resistivity are removed. In particular, the compressible non-resistive system, corresponding to vanishing magnetic diffusivity, is identified as a setting in which the existence theory remains unresolved on the whole space.

This question is distinct from the periodic, background-field problem addressed by the paper, because it concerns global small solutions on R3\mathbb{R}^3 without imposing any further structural condition on the system or the data.

References

For instance, whether the compressible non-resistive system ($\nu=0$) possesses global small solutions on $\mathbb R3$, without any further structural assumption, is still unknown.

— 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 “The model”