Dimension of singularities to the 3d simplified nematic liquid crystal flows. (December 2018) - Liu, Qiao
Abstract: 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, ∇ d ) ∈ L w ( 0, T ; L s ( R 3 ) ) with 3 < w, s < ∞, 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, ∇ d ) ∈ L ∞ ( 0, T ; L 3, ∞ ( R 3 ) ), where L 3, ∞ ( R 3 ) denotes the standard weak L 3 -Lebesgue space, then the parabolic upper Minkowski dimension of the singular set is bounded by 1.