On the Hollman McKenna conjecture: Interior concentration near curves
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,
Institute of Mathematics, Fudan University, Shanghai
School of Mathematic Sciences, Fudan University/Shanghai University of Medicine and Health Sciences, Shanghai, China
School of Mathematic Sciences, Soochow University, Suzhou, China
School of Mathematic Sciences, Fudan University, Shanghai, China
Received August 2015 Revised November 2015 Published July 2016
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 & Continuous Dynamical Systems - A, 2016, 36 (10) : 5579-5594. doi: 10.3934/dcds.2016045
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. doi: 10.1002/cpa.3160350604. Google Scholar
B. Chow, P. Lu and L. Ni, Hamilton's Ricci Flow, volume 77 of Graduate Studies in Mathematics,, American Mathematical Society, (2006). doi: 10.1090/gsm/077. Google Scholar
L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics,, American Mathematical Society, (1998). Google Scholar
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. doi: 10.1007/BF00281533. Google Scholar
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. doi: 10.1016/0022-0396(86)90096-3. Google Scholar
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics. Springer-Verlag, (2001). Google Scholar
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. doi: 10.1007/s00205-016-0983-1. Google Scholar
T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows,, Communications in Partial Differential Equations, 37 (2012), 875. doi: 10.1080/03605302.2012.659366. Google Scholar
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. doi: 10.1070/RM2003v058n02ABEH000609. Google Scholar
F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501. doi: 10.1002/cpa.3160480503. Google Scholar
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. Google Scholar
F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Ration. Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. Google Scholar
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187. Google Scholar
M. Struwe, On partial regularity results for the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 437. doi: 10.1002/cpa.3160410404. Google Scholar
W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations,, Aspects of Mathematics, (1985). doi: 10.1007/978-3-663-13911-9. Google Scholar
Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591
Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155
Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499
Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1
Jishan Fan, Tohru Ozawa. Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 859-867. doi: 10.3934/dcds.2009.25.859
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 & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems & Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873
Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475
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 & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure & Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407
Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. Kinetic & Related Models, 2012, 5 (3) : 505-516. doi: 10.3934/krm.2012.5.505
Apala Majumdar. The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1303-1337. doi: 10.3934/cpaa.2012.11.1303
Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems & Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623
Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. II. Kinetic & Related Models, 2014, 7 (2) : 291-304. doi: 10.3934/krm.2014.7.291
Jishan Fan, Tohru Ozawa. Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model. Kinetic & Related Models, 2009, 2 (2) : 293-305. doi: 10.3934/krm.2009.2.293
Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems & Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939
Xian-Gao Liu Jianzhong Min Kui Wang Xiaotao Zhang