Serrin's regularity results for the incompressible liquid crystals system

Previous Article
On the Hollman McKenna conjecture: Interior concentration near curves
DCDS Home
This Issue
Next Article
Deterministically driven random walks in a random environment on $\mathbb{Z}$

October 2016, 36(10): 5579-5594. doi: 10.3934/dcds.2016045

Serrin's regularity results for the incompressible liquid crystals system

Xian-Gao Liu ^1, , Jianzhong Min ^2, , Kui Wang ^3, and Xiaotao Zhang ^4,

1.	Institute of Mathematics, Fudan University, Shanghai
2.	School of Mathematic Sciences, Fudan University/Shanghai University of Medicine and Health Sciences, Shanghai, China
3.	School of Mathematic Sciences, Soochow University, Suzhou, China
4.	School of Mathematic Sciences, Fudan University, Shanghai, China

Received August 2015 Revised November 2015 Published July 2016

Abstract
Related Papers

In this paper, we study the simplified system of the original Ericksen--Leslie equations for the flow of liquid crystals [10]. Under Serrin criteria [13], we prove a partial interior regularity result of weak solutions for the three-dimensional incompressible liquid crystal system.

Keywords: partial regularity, Serrin criteria, heat kernel theory., Incompressible liquid crystals, Calderón-Zygmund inequality.

Mathematics Subject Classification: Primary: 35Q35; Secondary: 76A1.

Citation: Xian-Gao Liu, Jianzhong Min, Kui Wang, Xiaotao Zhang. Serrin's regularity results for the incompressible liquid crystals system. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5579-5594. doi: 10.3934/dcds.2016045

References:

[1]	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.
[2]	B. Chow, P. Lu and L. Ni, Hamilton's Ricci Flow, volume 77 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI; Science Press, New York, 2006. doi: 10.1090/gsm/077.
[3]	L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.
[4]	E. B. Fabes, B. F. Jones and N. M. Rivière, The initial value problem for the Navier-Stokes equations with data in $L^p$, Arch. Rational Mech. Anal., 45 (1972), 222-240. doi: 10.1007/BF00281533.
[5]	Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.
[6]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
[7]	T. Huang, F. Lin, C. Liu and C. Wang, Finite time singularity of the nematic liquid crystal flow in dimension three, Archive for Rational Mechanics and Analysis, (2016), 1-32, arXiv:1504.0108. doi: 10.1007/s00205-016-0983-1.
[8]	T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows, Communications in Partial Differential Equations, 37 (2012), 875-884. doi: 10.1080/03605302.2012.659366.
[9]	L. Iskauriaza, G. A. Serëgin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44. doi: 10.1070/RM2003v058n02ABEH000609.
[10]	F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.
[11]	F.-H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1996), 1-22.
[12]	F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.
[13]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.
[14]	M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404.
[15]	W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Aspects of Mathematics, E8. Friedr. Vieweg & Sohn, Braunschweig, 1985. doi: 10.1007/978-3-663-13911-9.

show all references

References:

[1]	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.
[2]	B. Chow, P. Lu and L. Ni, Hamilton's Ricci Flow, volume 77 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI; Science Press, New York, 2006. doi: 10.1090/gsm/077.
[3]	L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.
[4]	E. B. Fabes, B. F. Jones and N. M. Rivière, The initial value problem for the Navier-Stokes equations with data in $L^p$, Arch. Rational Mech. Anal., 45 (1972), 222-240. doi: 10.1007/BF00281533.
[5]	Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.
[6]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
[7]	T. Huang, F. Lin, C. Liu and C. Wang, Finite time singularity of the nematic liquid crystal flow in dimension three, Archive for Rational Mechanics and Analysis, (2016), 1-32, arXiv:1504.0108. doi: 10.1007/s00205-016-0983-1.
[8]	T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows, Communications in Partial Differential Equations, 37 (2012), 875-884. doi: 10.1080/03605302.2012.659366.
[9]	L. Iskauriaza, G. A. Serëgin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44. doi: 10.1070/RM2003v058n02ABEH000609.
[10]	F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.
[11]	F.-H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1996), 1-22.
[12]	F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.
[13]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.
[14]	M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404.
[15]	W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Aspects of Mathematics, E8. Friedr. Vieweg & Sohn, Braunschweig, 1985. doi: 10.1007/978-3-663-13911-9.

[1]	Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
[2]	Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591
[3]	Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155
[4]	Sun-Sig Byun, Yumi Cho, Shuang Liang. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3843-3855. doi: 10.3934/dcdsb.2020038
[5]	Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499
[6]	Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1
[7]	Jishan Fan, Tohru Ozawa. Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 859-867. doi: 10.3934/dcds.2009.25.859
[8]	Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure and Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[9]	Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[10]	Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873
[11]	Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475
[12]	Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems and Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
[13]	Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
[14]	Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure and Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407
[15]	Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. Kinetic and Related Models, 2012, 5 (3) : 505-516. doi: 10.3934/krm.2012.5.505
[16]	Apala Majumdar. The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1303-1337. doi: 10.3934/cpaa.2012.11.1303
[17]	María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems and Imaging, 2022, 16 (1) : 251-281. doi: 10.3934/ipi.2021049
[18]	Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623
[19]	Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems and Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
[20]	Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. II. Kinetic and Related Models, 2014, 7 (2) : 291-304. doi: 10.3934/krm.2014.7.291

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Xian-Gao Liu Jianzhong Min Kui Wang Xiaotao Zhang
on Google Scholar
Xian-Gao Liu Jianzhong Min Kui Wang Xiaotao Zhang

[Back to Top]