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September  2021, 41(9): 4397-4419. 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  September 2021 Early access  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<r\leq 1 $
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 and Continuous Dynamical Systems, 2021, 41 (9) : 4397-4419. doi: 10.3934/dcds.2021041
References:
[1]

L. CaffarelliR. 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.

[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.

[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.

[4]

J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.

[5]

L. EscauriazaG. 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.

[6]

K. Falconer, Fractal Geometry, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014.

[7]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, 2003. doi: 10.1142/9789812795557.

[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.

[9]

S. GustafsonK. 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.

[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.

[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.

[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.

[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.

[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.

[15]

F. Leslie, Theory of flow phenomenum in liquid crystals., The Theory of Liquid Crystals, London-New York: Academic Press, 4 (1979), 1-81. 

[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.

[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.

[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.

[19]

F. LinJ. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

W. RenY. 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.

[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.

[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.

[29]

V. Scheffer, Hausforff measure and the the Navier–Stokes equations, Commun. Math. Phys., 55 (1977), 97-112.  doi: 10.1007/BF01626512.

[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.

[31]

I. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals, Taylor & Francis, London, New York, 2004.

[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.

[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.

[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.

[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.

show all references

References:
[1]

L. CaffarelliR. 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.

[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.

[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.

[4]

J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.

[5]

L. EscauriazaG. 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.

[6]

K. Falconer, Fractal Geometry, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014.

[7]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, 2003. doi: 10.1142/9789812795557.

[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.

[9]

S. GustafsonK. 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.

[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.

[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.

[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.

[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.

[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.

[15]

F. Leslie, Theory of flow phenomenum in liquid crystals., The Theory of Liquid Crystals, London-New York: Academic Press, 4 (1979), 1-81. 

[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.

[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.

[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.

[19]

F. LinJ. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

W. RenY. 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.

[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.

[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.

[29]

V. Scheffer, Hausforff measure and the the Navier–Stokes equations, Commun. Math. Phys., 55 (1977), 97-112.  doi: 10.1007/BF01626512.

[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.

[31]

I. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals, Taylor & Francis, London, New York, 2004.

[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.

[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.

[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.

[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.

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