PARTIAL REGULARITY AND THE MINKOWSKI DIMENSION OF SINGULAR POINTS FOR SUITABLE WEAK SOLUTIONS TO THE 3D SIMPLIFIED ERICKSEN-LESLIE SYSTEM

We study the partial regularity problem for a three dimensional simplified Ericksen-Leslie system, which consists of the Navier-Stokes equations for the fluid velocity coupled with a convective Ginzburg-Landau type equations for the molecule orientation, modelling the incompressible nematic liquid crystal flows. Base on the recent studies on the Navier-Stokes equations, we first prove some new local energy bounds and an epsilon-regularity criterion for suitable weak solutions to the simplified Ericksen-Leslie system, i.e., for sigma is an element of [0, 1], there exists a epsilon > 0 such that if (u, d, P) is a suitable weak solution in Q(r)(z(0)) with 0 < r <= 1 and z(0) = (x(0), t(0)), and satisfies r(-3/2-sigma) integral(t0)(t0-r2) (parallel to vertical bar u vertical bar(2)parallel to(2/2-sigma)(H-sigma(Br(x0))) + parallel to vertical bar del d vertical bar(2)parallel to(2/2-sigma)(H-sigma(Br(x0))) + parallel to P parallel to(2/2-sigma)(H-sigma(Br(x0))))dt <= epsilon, then (u, d) is regular at z(0). Here, H-sigma (B-r(x)) is the dual space of H-0(sigma) (B-r(x)), the space of functions f in the homogeneous Sobolev space (H) over dot(sigma)(R-3) such that supp f subset of <(B-r(x))over bar>. Inspired by this epsilon-regularity criterion, we then improve the known upper Minkowski dimension of the possible interior singular points for suitable weak solutions from 95/63 (approximate to 1.50794) given by [24] (Nonlinear Anal. RWA, 44 (2018), 246-259.) to 835/613 (approximate to 1.36215).