Multidimensional compressible NSCH with vacuum beyond local well-posedness

Establish the corresponding multidimensional theory for the compressible Navier–Stokes/Cahn–Hilliard system with vacuum, extending the one-dimensional global existence and uniqueness result and addressing the unresolved multidimensional problem.

Background

The paper studies the three-dimensional compressible Navier–Stokes/Cahn–Hilliard system when the initial density may vanish. Before this work, global existence and uniqueness of strong solutions allowing vacuum had been established only in one spatial dimension by Ding, Li, Yan, and Zheng. The multidimensional counterpart had been raised previously for related incompressible nonhomogeneous and compressible models.

The paper proves local well-posedness of strong solutions in three dimensions with initial vacuum, but it does not resolve the broader multidimensional problem referenced in the introduction. In particular, the cited open problem concerns extending the available vacuum theory to the multidimensional compressible setting, beyond the local result established here.

References

The corresponding problem in the multidimensional case, raised in for the incompressible nonhomogeneous model and in for the compressible one, remains open.

— Local Existence and Uniqueness for the 3D Compressible Navier-Stokes/Cahn-Hilliard System with Vacuum  (2610.11193 - Li et al., 8 Oct 2026) in Section 1, Introduction, paragraph beginning “When vacuum is taken into account in compressible two-phase flows”