Dimension of singularities to the 3d simplified nematic liquid crystal flows

In this paper, we study the possible singular set of suitable weak solutions (u,d) to the 3d simplified nematic liquid crystal flows, and prove that the parabolic upper Minkowski dimension of the possible singular set is bounded by 9563. Moreover, if the suitable weak solution (u,d) of the 3d simplified nematic liquid crystal flows satisfies (u,∇d)∈Lw(0,T;Ls(R3))with3