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 95/63. Moreover, if the suitable weak solution (u, d) of the 3d simplified nematic liquid crystal flows satisfies (u,del d) is an element of L-w (0,T; L-s (R-3)) with3 < w,s < infinity, then the parabolic upper Minkowski dimension of the singular set is no greater than max{w, s} (2/w + 3/s - 1). If the suitable weak solution (u, d) satisfies (u, del d) is an element of L-infinity (0,T; L-3,L-infinity (R-3)), where L-3,L-infinity(R-3) denotes the standard weak L-3-Lebesgue space, then the parabolic upper Minkowski dimension of the singular set is bounded by 1. (C) 2018 Elsevier Ltd. All rights reserved.