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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

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, 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.  Google Scholar

[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.  Google Scholar

[3]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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