Dimension of singularities to the 3d simplified nematic liquid crystal flows. (December 2018)

• Record Type: Journal Article
• Authors: Liu, Qiao
• Is Part Of: Nonlinear analysis. Volume 44(2018)
• Page Start: 246
• Page End: 259
• Publication Date: 2018-12
• Subjects: Nematic liquid crystal flows -- Singular point -- Hausdorff dimension -- Minkowski dimension; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2018.05.005
• Languages: English
• ISSNs: 1468-1218

Abstract

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.