Parabolic fractal dimension of forward-singularities for Navier-Stokes and liquid crystals inequalities

In 1985, V. Scheffer discussed partial regularity for what he called solutions to the "Navier-Stokes inequality", which only satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system. One may extend this notion to a system introduced by F.-H. Lin and C. Liu in 1995 to model the flow of nematic liquid crystals, which include the Navier-Stokes system when the 'director field' $ d $ is taken to be zero. The model includes a further parabolic system which implies an a priori maximum principle for $ d $, which is lost when one considers the analogous 'inequality'.

In 2018, Q. Liu proved a partial regularity result for solutions to the Lin-Liu model in terms of the 'parabolic fractal dimension' $ \text{dim}_{ \text{pf}} $, relying on the boundedness of $ d $ coming from the maximum principle. Q. Liu proves $ { \text{dim}_{ \text{pf}}(\Sigma_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}} $ for any compact $ \mathcal{K} $, where $ \Sigma_{-} $ is the set of space-time points near which the solution blows up forwards in time. For solutions to the corresponding 'inequality', we prove that, without any compensation for the lack of maximum principle, one has $ { \text{dim}_{ \text{pf}}(\Sigma_{-} \cap \mathcal{K}) \leq \tfrac {55}{13}} $. We also provide a range of criteria, including as just one example the boundedness of $ d $, any one of which would furthermore imply that solutions to the inequality also satisfy $ { \text{dim}_{ \text{pf}}(\Sigma_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}} $.