Removal of geometric director constraints in incompressible liquid-crystal flow

Determine whether the hemisphere or angular constraints on the initial director field can be removed for the incompressible liquid-crystal flow in three-dimensional Euclidean space.

Background

The paper discusses the role of hemisphere and angle conditions on the initial director field in preventing concentration phenomena and enabling strong convergence of director gradients. It states that removing these geometric restrictions remains unresolved even for the incompressible model in three-dimensional space, making this a broader open problem related to the director equation.

References

Nevertheless, removing such geometric constraints remains open even for the incompressible model in $\mathbb{R}3 $.

— 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