Well-posedness of the co-rotational Beris-Edwards system with the Landau-De Gennes energy in $ L^{3}_{uloc}(\mathbb{R}^3) $

Qiao Liu; Jihong Zhao

doi:10.3934/dcds.2024081

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2024, Volume 44, Issue 12: 3878-3919. Doi: 10.3934/dcds.2024081

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Well-posedness of the co-rotational Beris-Edwards system with the Landau-De Gennes energy in $ L^{3}_{uloc}(\mathbb{R}^3) $

Qiao Liu^1, , and
Jihong Zhao^2,

1.
School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083 Hunan, China
2.
School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, 721013 Shaanxi, China

^*Corresponding author: Qiao Liu
^*Corresponding author: Qiao Liu

Received: November 2023

Revised: April 2024

Early access: June 2024

Published: December 2024

Q. Liu is supported by the National Natural Science Foundation of China (12071122) and the Natural Science Foundation of Hunan Province (2023JJ10059). J. Zhao is partially supported by the National Natural Science Foundation of China (no. 12361034) and the Natural Science Foundation of Shaanxi Province (no. 2022JM-034).

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Abstract

We investigate the well-posedness of the 3d co-rotational Beris-Edwards system for incompressible nematic liquid crystal flows with the Landau-De Gennes bulk potential. The system under consideration consists of the Navier–Stokes equations for the fluid velocity $ \boldsymbol u $, and an evolution equation for the $ Q $-tensor order parameter. We prove the existence of solutions to the Cauchy problem of the system with initial data $ ( \boldsymbol u_0,Q_0) $ having small $ L_{uloc}^3(\mathbb{R}^3) $-norm of $ ( \boldsymbol u_0,\nabla Q_0) $. Moreover, the uniqueness of $ L^3_{uloc} $-solutions is obtained.

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Mathematics Subject Classification: 35Q35, 76D03, 76A15.

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References

[1]	H. Abels, G. Dolzmann and Y. Liu, Well-posedness of a fully coupled Navier–Stokes/Q-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal., 46 (2014), 3050-3077. doi: 10.1137/130945405.
[2]	H. Abels, G. Dolzmann and Y. Liu, Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions, Adv. Differential Equations, 21 (2016), 109-152. doi: 10.57262/ade/1448323166.
[3]	J. M. Ball, Mathematics of liquid crystal, Molecular Crystals and Liquid Crystals, 647 (2016), 1-27. doi: 10.1080/15421406.2017.1289425.
[4]	J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a continuum theory, Molecular Crystals and Liquid Crystals, 525 (2010), 1-11. doi: 10.1080/15421401003795555.
[5]	J. M. Ball and A. Zarnescu, Orientability and energy minimization in liquid crystal models, Arch. Rational Mech. Anal., 202 (2011), 493-535. doi: 10.1007/s00205-011-0421-3.
[6]	A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems: With Internal Microstructure, Oxford University Press, 1994. doi: 10.1093/oso/9780195076943.001.0001.
[7]	L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604.
[8]	C. Cavaterra, E. Rocca, H. Wu and X. Xu, Global strong solutions of the full Navier–Stokes and Q-tensor system for nematic liquid crystal flows in two dimensions, SIAM J. Math. Anal., 48 (2016), 1368-1399. doi: 10.1137/15M1048550.
[9]	G.-Q. Chen, A. Majumdar, D. Wang and R. Zhang, Global existence and regularity of solutions for active liquid crystals, J. Differential Equations, 263 (2017), 202-239. doi: 10.1016/j.jde.2017.02.035.
[10]	M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. E. Schonbek, On asymptotic isotropy for a hydrodynamic model of liquid crystals, Asymptot. Anal., 97 (2016), 189-210. doi: 10.3233/ASY-151348.
[11]	P. G. De Gennes and J. Prost, The Physics of Liquid Crystals, 2nd edition, Oxford University Press, 1995. doi: 10.1093/oso/9780198520245.001.0001.
[12]	H. Du, X. Hu and C. Wang, Suitable weak solutions for the co-rotational Beris-Edwards system in dimension three, Arch. Rational Mech. Anal., 238 (2020), 749-803. doi: 10.1007/s00205-020-01554-y.
[13]	J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392.
[14]	F. Guillen-Gonzalez and M. A. Rodriguez-Bellido, A uniqueness and regularity criterion for Q-tensor models with Neumann boundary conditions, Differential Integral Equations, 28 (2015), 537-552. doi: 10.57262/die/1427744100.
[15]	F. Guillen-Gonzalez and M. A. Rodriguez-Bellido, Weak time regularity and uniqueness for a Q-tensor model, SIAM J. Math. Anal., 46 (2014), 3540-3567. doi: 10.1137/13095015X.
[16]	J. L. Hineman and C. Wang, Well-posedness of nematic liquid crystal flow in $L^3_{uloc}(\mathbb{R}^3)$, Arch. Rational Mech. Anal., 210 (2013), 177-218. doi: 10.1007/s00205-013-0643-7.
[17]	M.-C. Hong and Y. Mei, Well-posedness of the Ericksen-Leslie system with the Oseen-Frank energy in $L^3_{uloc}(\mathbb{R}^3)$, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 3, 38 pp. doi: 10.1007/s00526-018-1453-4.
[18]	J. Huang and S. Ding, Global well-posedness for the dynamical Q-tensor model of liquid crystals, Sci. China Math., 58 (2015), 1349-1366. doi: 10.1007/s11425-015-4990-8.
[19]	G. Iyer, X. Xu and A. D. Zarnescu, Dynamic cubic instability in a 2D Q-tensor model for liquid crystals, Math. Models Methods Appl. Sci., 25 (2015), 1477-1517. doi: 10.1142/S0218202515500396.
[20]	P. G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, Chapman & Hall/CRC Res. Notes Math., Vol 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.
[21]	F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.
[22]	J. Li, E. S. Titi and Z. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb{R}^2$, Math. Models Methods Appl. Sci., 26 (2016), 803-822. doi: 10.1142/S0218202516500184.
[23]	F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.
[24]	F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571. doi: 10.1002/cpa.21583.
[25]	Q. Liu, C. Wang, X. Zhang and J. Zhou, On optimal boundary control of Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 38, 64 pp. doi: 10.1007/s00526-019-1676-z.
[26]	A. Majumdar and A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Rational Mech. Anal., 196 (2010), 227-280. doi: 10.1007/s00205-009-0249-2.
[27]	N. J. Mottram and C. J. P. Newton, Introduction to $Q$-tensor Theory, arXiv: 1409.3542, 2014.
[28]	M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system, Arch. Rational Mech. Anal., 203 (2012), 45-67. doi: 10.1007/s00205-011-0443-x.
[29]	M. Paicu and A. Zarnescu, Global existence and regularity for the full coupled Navier-Stokes and Q-tensor system, SIAM J. Math. Anal., 43 (2011), 2009-2049. doi: 10.1137/10079224X.
[30]	G. Seregin and V. Šverśk, On global weak solutions to the Cauchy problem for the Navier–Stokes equations with large $L_3$-initial data, Nonlinear Anal., 154 (2017), 269-296. doi: 10.1016/j.na.2016.01.018.
[31]	E. Stein, Singular Integrals and Differentiabiblity Properties of Functions, Princeton University Press, Princeton, 1970.
[32]	W. Wang, P. Zhang and Z. Zhang, Rigorous derivation from Landau-De Gennes theory to Ericksen-Leslie theory, SIAM J. Math. Anal., 47 (2015), 127-158. doi: 10.1137/13093529X.
[33]	M. Wilkinson, Strictly physical global weak solutions of a Navier–Stokes Q-tensor system with singular potential, Arch. Rational Mech. Anal., 218 (2015), 487-526. doi: 10.1007/s00205-015-0864-z.
[34]	H. Wu, X. Xu and A. Zarnescu, Dynamics and flow effects in the Beris-Edwards system modeling nematic liquid crystals, Arch. Rational Mech. Anal., 231 (2019), 1217-1267. doi: 10.1007/s00205-018-1297-2.
[35]	Y. Xiao, Global strong solution to the three-dimensional liquid crystal flows of $Q$-tensor model, J. Differential Equations, 262 (2017), 1291-1316. doi: 10.1016/j.jde.2016.10.011.
[36]	A. Zarnescu, Topics in the Q-tensor theory of liquid crystals, Topics in Mathematical Modeling and Analysis, 187-252, Jindřich Nečas Cent. Math. Model. Lect. Notes, 7, Matfyzpress, Prague, 2012.