# American Institute of Mathematical Sciences

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

## Serrin's regularity results for the incompressible liquid crystals system

 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

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.
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.

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##### 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.
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