# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021041

## Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system

 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Received  August 2020 Published  March 2021

Fund Project: Q. Liu is supported by the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 19A313), and the National Natural Science Foundation of China (No. 12071122)

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
 $\varepsilon$
-regularity criterion for suitable weak solutions to the simplified Ericksen-Leslie system, i.e., for
 $\sigma\in [0,1]$
, there exists a
 $\varepsilon>0$
such that if
 $(u,d,P)$
is a suitable weak solution in
 $Q_{r}(z_{0})$
with
 $0 and $ z_{0} = (x_{0},t_{0}) $, and satisfies $ \begin{align*} r^{-\frac{3}{2-\sigma}}\!\!\int_{\!t_{0}-r^{2}}^{t_{0}}\! (\||u|^{2}\|_{\!H^{-\sigma}(B_{r}(x_{0}))}^{\frac{2}{2-\sigma}} \!+\!\||\nabla d|^{2}\|_{\!H^{-\sigma}(B_{r}(x_{0}))}^{\frac{2}{2-\sigma}} \!+\!\|P\|_{\!H^{-\sigma}(B_{r}(x_{0}))}^{\frac{2}{2-\sigma}})\text{d}t\leq \varepsilon, \end{align*} $then $ (u, d) $is regular at $ z_{0} $. Here, $ H^{-\sigma}(B_{r}(x)) $is the dual space of $ H^{\sigma}_{0}(B_{r}(x)) $, the space of functions $ f $in the homogeneous Sobolev space $ \dot{H}^{\sigma}(\mathbb{R}^{3}) $such that $ \operatorname{supp} f\subset \overline{B_{r}(x)} $. Inspired by this $ \varepsilon $-regularity criterion, we then improve the known upper Minkowski dimension of the possible interior singular points for suitable weak solutions from $ \frac{95}{63} (\approx 1.50794) $given by [24] (Nonlinear Anal. RWA, 44 (2018), 246–259.) to $ \frac{835}{613} (\approx 1.36215) $. Citation: Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021041 ##### References:  [1] L. Caffarelli, R. Kohn and L. Nireberg, Partial regularity of suitable weak solutions of Navier–Stokes equations, Commun. Partial Differential Equations, 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. Google Scholar [2] H. Choe and J. Lewis, On the singular set in the Navier–Stokes equations, J. Funct. Anal., 175 (2000), 348-369. doi: 10.1006/jfan.2000.3582. Google Scholar [3] M. Dai and M. Schonbek, Asymptotic behavior of solutions to the liquid crystal system in$H^{m}(\mathbb{R}^{3})$, SIAM J. Math. Anal., 46 (2014), 3131-3150. doi: 10.1137/120895342. Google Scholar [4] J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358. Google Scholar [5] L. Escauriaza, G. Seregin and V.$\breve{\mathrm{S}}$verák,$L^{3, \infty}$solutions to the Navier–Stokes equations and backward uniqueness, Russian Math. Surveys, 58 (2003), 211-250. doi: 10.1070/RM2003v058n02ABEH000609. Google Scholar [6] K. Falconer, Fractal Geometry, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014. Google Scholar [7] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, 2003. doi: 10.1142/9789812795557. Google Scholar [8] C. Guevara and N. Phuc, Local energy bounds and$\varepsilon$-regularity criteria for the 3D Navier–Stokes system, Calc. Var., 56 (2017), Paper No. 68, 16 pp. doi: 10.1007/s00526-017-1151-7. Google Scholar [9] S. Gustafson, K. Kang and T. Tsai, Interior regularity criteria for suitable weak solutions of the Navier–Stokes equations, Commun. Math. Phys., 273 (2007), 161-176. doi: 10.1007/s00220-007-0214-6. Google Scholar [10] Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997. Google Scholar [11] X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Commum. Math. Phys., 296 (2010), 861-880. doi: 10.1007/s00220-010-1017-8. Google Scholar [12] Y. Koh and M. Yang, The Minkowski dimension of interior singular points in the incompressible Navier–Stokes equations, J. Differential Equations, 261 (2016), 3137-3148. doi: 10.1016/j.jde.2016.05.020. Google Scholar [13] I. Kukavica, The fractal dimension of the singular set for solutions of the Navier–Stokes system, Nonlinearity, 22 (2009), 2889-2900. doi: 10.1088/0951-7715/22/12/005. Google Scholar [14] B. Lai and W. Ma, On the interior regularity criteria for liquid crystal flows, Nonlinear Anal. Real Word Appl., 40 (2018), 1-13. doi: 10.1016/j.nonrwa.2017.08.006. Google Scholar [15] F. Leslie, Theory of flow phenomenum in liquid crystals., The Theory of Liquid Crystals, London-New York: Academic Press, 4 (1979), 1-81. Google Scholar [16] X. Li and D. Wang, Global solution to the incompressible flow of liquid crystal, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045. Google Scholar [17] F. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Commun. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. Google Scholar [18] F. Lin, A new proof of Caffarelli-Kohn-Nirenberg theorem, Commun. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. Google Scholar [19] F. Lin, J. Lin and C. Wang, Liquid crystal flow in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. Google Scholar [20] F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503. Google Scholar [21] F. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-22. doi: 10.3934/dcds.1996.2.1. Google Scholar [22] F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Annal. Math. Ser. B, 31 (2010), 921-938. doi: 10.1007/s11401-010-0612-5. Google Scholar [23] F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Commun. Pure Appl. Math., 69 (2016), 1532-1571. doi: 10.1002/cpa.21583. Google Scholar [24] Q. Liu, Dimension of singularities to the 3d simplified nematic liquid crystal flows, Nonlinear Analysis: Real World Applications, 44 (2018), 246-259. doi: 10.1016/j.nonrwa.2018.05.005. Google Scholar [25] Q. Liu, Regularity of weak solutions and the number of singular points to the 3d simplified nematic liquid crystal system, J. Funct. Anal., 277 (2019), 108294, 33 pp. doi: 10.1016/j.jfa.2019.108294. Google Scholar [26] W. Ren, Y. Wang and G. Wu, Remarks on the singular set of suitable weak solutions for the three-dimensional Navier–Stokes equations, J. Math. Anal. Appl., 467 (2018), 807-824. doi: 10.1016/j.jmaa.2018.07.003. Google Scholar [27] J. Robinson and W. Sadowski, Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier–Stokes equations, Nonlinearity, 22 (2009), 2093-2099. doi: 10.1088/0951-7715/22/9/002. Google Scholar [28] V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations, Pac. J. Math., 66 (1976), 535-552. doi: 10.2140/pjm.1976.66.535. Google Scholar [29] V. Scheffer, Hausforff measure and the the Navier–Stokes equations, Commun. Math. Phys., 55 (1977), 97-112. doi: 10.1007/BF01626512. Google Scholar [30] G. Seregin, On the number of singular points of weak solutions to the Navier–Stokes equations, Commun. Pure Appl. Math., 54 (2001), 1019-1028. doi: 10.1002/cpa.3002. Google Scholar [31] I. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals, Taylor & Francis, London, New York, 2004. Google Scholar [32] Y. Wang and G. Wu, On the box-counting dimension of potential singular set for suitable weak solutions to the 3D Navier–Stokes equations, Nonlinearity, 30 (2017), 1762-1772. doi: 10.1088/1361-6544/aa6444. Google Scholar [33] Y. Wang and M. Yang, Improved bounds for box dimensions of potential singular points to the Navier–Stokes equations, Nonlinearity, 32 (2019), 4817–4833, arXiv: 1812.00900v1 [math.AP]. doi: 10.1088/1361-6544/ab3f51. Google Scholar [34] H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379. Google Scholar [35] X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181. doi: 10.1016/j.jde.2011.08.028. Google Scholar show all references ##### References:  [1] L. Caffarelli, R. Kohn and L. Nireberg, Partial regularity of suitable weak solutions of Navier–Stokes equations, Commun. Partial Differential Equations, 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. Google Scholar [2] H. Choe and J. Lewis, On the singular set in the Navier–Stokes equations, J. Funct. Anal., 175 (2000), 348-369. doi: 10.1006/jfan.2000.3582. Google Scholar [3] M. Dai and M. Schonbek, Asymptotic behavior of solutions to the liquid crystal system in$H^{m}(\mathbb{R}^{3})$, SIAM J. Math. Anal., 46 (2014), 3131-3150. doi: 10.1137/120895342. Google Scholar [4] J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358. Google Scholar [5] L. Escauriaza, G. Seregin and V.$\breve{\mathrm{S}}$verák,$L^{3, \infty}$solutions to the Navier–Stokes equations and backward uniqueness, Russian Math. Surveys, 58 (2003), 211-250. doi: 10.1070/RM2003v058n02ABEH000609. Google Scholar [6] K. Falconer, Fractal Geometry, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014. Google Scholar [7] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, 2003. doi: 10.1142/9789812795557. Google Scholar [8] C. Guevara and N. Phuc, Local energy bounds and$\varepsilon$-regularity criteria for the 3D Navier–Stokes system, Calc. Var., 56 (2017), Paper No. 68, 16 pp. doi: 10.1007/s00526-017-1151-7. Google Scholar [9] S. Gustafson, K. Kang and T. Tsai, Interior regularity criteria for suitable weak solutions of the Navier–Stokes equations, Commun. Math. Phys., 273 (2007), 161-176. doi: 10.1007/s00220-007-0214-6. Google Scholar [10] Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997. Google Scholar [11] X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Commum. Math. Phys., 296 (2010), 861-880. doi: 10.1007/s00220-010-1017-8. Google Scholar [12] Y. Koh and M. Yang, The Minkowski dimension of interior singular points in the incompressible Navier–Stokes equations, J. Differential Equations, 261 (2016), 3137-3148. doi: 10.1016/j.jde.2016.05.020. Google Scholar [13] I. Kukavica, The fractal dimension of the singular set for solutions of the Navier–Stokes system, Nonlinearity, 22 (2009), 2889-2900. doi: 10.1088/0951-7715/22/12/005. Google Scholar [14] B. Lai and W. Ma, On the interior regularity criteria for liquid crystal flows, Nonlinear Anal. Real Word Appl., 40 (2018), 1-13. doi: 10.1016/j.nonrwa.2017.08.006. Google Scholar [15] F. Leslie, Theory of flow phenomenum in liquid crystals., The Theory of Liquid Crystals, London-New York: Academic Press, 4 (1979), 1-81. Google Scholar [16] X. Li and D. Wang, Global solution to the incompressible flow of liquid crystal, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045. Google Scholar [17] F. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Commun. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. Google Scholar [18] F. Lin, A new proof of Caffarelli-Kohn-Nirenberg theorem, Commun. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. Google Scholar [19] F. Lin, J. Lin and C. Wang, Liquid crystal flow in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. Google Scholar [20] F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503. Google Scholar [21] F. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-22. doi: 10.3934/dcds.1996.2.1. Google Scholar [22] F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Annal. Math. Ser. B, 31 (2010), 921-938. doi: 10.1007/s11401-010-0612-5. Google Scholar [23] F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Commun. Pure Appl. Math., 69 (2016), 1532-1571. doi: 10.1002/cpa.21583. Google Scholar [24] Q. Liu, Dimension of singularities to the 3d simplified nematic liquid crystal flows, Nonlinear Analysis: Real World Applications, 44 (2018), 246-259. doi: 10.1016/j.nonrwa.2018.05.005. Google Scholar [25] Q. Liu, Regularity of weak solutions and the number of singular points to the 3d simplified nematic liquid crystal system, J. Funct. Anal., 277 (2019), 108294, 33 pp. doi: 10.1016/j.jfa.2019.108294. Google Scholar [26] W. Ren, Y. Wang and G. Wu, Remarks on the singular set of suitable weak solutions for the three-dimensional Navier–Stokes equations, J. Math. Anal. Appl., 467 (2018), 807-824. doi: 10.1016/j.jmaa.2018.07.003. Google Scholar [27] J. Robinson and W. Sadowski, Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier–Stokes equations, Nonlinearity, 22 (2009), 2093-2099. doi: 10.1088/0951-7715/22/9/002. Google Scholar [28] V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations, Pac. J. Math., 66 (1976), 535-552. doi: 10.2140/pjm.1976.66.535. Google Scholar [29] V. Scheffer, Hausforff measure and the the Navier–Stokes equations, Commun. Math. Phys., 55 (1977), 97-112. doi: 10.1007/BF01626512. Google Scholar [30] G. Seregin, On the number of singular points of weak solutions to the Navier–Stokes equations, Commun. Pure Appl. Math., 54 (2001), 1019-1028. doi: 10.1002/cpa.3002. Google Scholar [31] I. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals, Taylor & Francis, London, New York, 2004. Google Scholar [32] Y. Wang and G. Wu, On the box-counting dimension of potential singular set for suitable weak solutions to the 3D Navier–Stokes equations, Nonlinearity, 30 (2017), 1762-1772. doi: 10.1088/1361-6544/aa6444. Google Scholar [33] Y. Wang and M. Yang, Improved bounds for box dimensions of potential singular points to the Navier–Stokes equations, Nonlinearity, 32 (2019), 4817–4833, arXiv: 1812.00900v1 [math.AP]. doi: 10.1088/1361-6544/ab3f51. Google Scholar [34] H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379. Google Scholar [35] X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181. doi: 10.1016/j.jde.2011.08.028. Google Scholar  [1] Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021069 [2] Qian Cao, Yongli Cai, Yong Luo. Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021095 [3] Tao Wang. Variational relations for metric mean dimension and rate distortion dimension. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021050 [4] Thomas Barthelmé, Andrey Gogolev. Centralizers of partially hyperbolic diffeomorphisms in dimension 3. 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