## Global strong solution to the density-dependent incompressible flow of liquid crystals

HTML articles powered by AMS MathViewer

- by Xiaoli Li and Dehua Wang PDF
- Trans. Amer. Math. Soc.
**367**(2015), 2301-2338 Request permission

## Abstract:

The initial-boundary value problem for the density-dependent incompressible flow of liquid crystals is studied in a three-dimensional bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness are established for both the local strong solution with large initial data and the global strong solution with ‘small’ data. It is also proved that when the strong solution exists, a weak solution with the same data must be equal to the unique strong solution.## References

- Herbert Amann,
*Linear and quasilinear parabolic problems. Vol. I*, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR**1345385**, DOI 10.1007/978-3-0348-9221-6 - Jöran Bergh and Jörgen Löfström,
*Interpolation spaces. An introduction*, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR**0482275**, DOI 10.1007/978-3-642-66451-9 - L. Caffarelli, R. Kohn, and L. Nirenberg,
*Partial regularity of suitable weak solutions of the Navier-Stokes equations*, Comm. Pure Appl. Math.**35**(1982), no. 6, 771–831. MR**673830**, DOI 10.1002/cpa.3160350604 - S. Chandrasekhar,
*Liquid crystals.*2nd ed., Cambridge University Press, 1992. - Blanca Climent-Ezquerra, Francisco Guillén-González, and Marko Rojas-Medar,
*Reproductivity for a nematic liquid crystal model*, Z. Angew. Math. Phys.**57**(2006), no. 6, 984–998. MR**2279252**, DOI 10.1007/s00033-005-0038-1 - Daniel Coutand and Steve Shkoller,
*Well-posedness of the full Ericksen-Leslie model of nematic liquid crystals*, C. R. Acad. Sci. Paris Sér. I Math.**333**(2001), no. 10, 919–924 (English, with English and French summaries). MR**1873808**, DOI 10.1016/S0764-4442(01)02161-9 - R. Danchin,
*Global existence in critical spaces for compressible Navier-Stokes equations*, Invent. Math.**141**(2000), no. 3, 579–614. MR**1779621**, DOI 10.1007/s002220000078 - R. Danchin,
*Density-dependent incompressible fluids in bounded domains*, J. Math. Fluid Mech.**8**(2006), no. 3, 333–381. MR**2258416**, DOI 10.1007/s00021-004-0147-1 - Benoit Desjardins,
*Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations*, Differential Integral Equations**10**(1997), no. 3, 577–586. MR**1744862** - Shijin Ding, Junyu Lin, Changyou Wang, and Huanyao Wen,
*Compressible hydrodynamic flow of liquid crystals in 1-D*, Discrete Contin. Dyn. Syst.**32**(2012), no. 2, 539–563. MR**2837072**, DOI 10.3934/dcds.2012.32.539 - Shijin Ding, Changyou Wang, and Huanyao Wen,
*Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one*, Discrete Contin. Dyn. Syst. Ser. B**15**(2011), no. 2, 357–371. MR**2754089**, DOI 10.3934/dcdsb.2011.15.357 - J. L. Ericksen,
*Hydrostatic theory of liquid crystals*, Arch. Rational Mech. Anal.**9**(1962), 371–378. MR**137403**, DOI 10.1007/BF00253358 - Benoît Desjardins,
*Regularity of weak solutions of the compressible isentropic Navier-Stokes equations*, Comm. Partial Differential Equations**22**(1997), no. 5-6, 977–1008. MR**1452175**, DOI 10.1080/03605309708821291 - J. L. Ericksen,
*Conservation laws for liquid crystals*, Trans. Soc. Rheol.**5**(1961), 23–34. MR**158610**, DOI 10.1122/1.548883 - J. L. Ericksen,
*Continuum theory of nematic liquid crystals.*Res. Mechanica**21**(1987), 381-392. - Hiroshi Fujita and Tosio Kato,
*On the Navier-Stokes initial value problem. I*, Arch. Rational Mech. Anal.**16**(1964), 269–315. MR**166499**, DOI 10.1007/BF00276188 - Giovanni P. Galdi,
*An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I*, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR**1284205**, DOI 10.1007/978-1-4612-5364-8 - Robert Hardt and David Kinderlehrer,
*Mathematical questions of liquid crystal theory*, Theory and applications of liquid crystals (Minneapolis, Minn., 1985) IMA Vol. Math. Appl., vol. 5, Springer, New York, 1987, pp. 151–184. MR**900833**, DOI 10.1007/978-1-4613-8743-5_{9} - Robert Hardt, David Kinderlehrer, and Fang-Hua Lin,
*Existence and partial regularity of static liquid crystal configurations*, Comm. Math. Phys.**105**(1986), no. 4, 547–570. MR**852090**, DOI 10.1007/BF01238933 - Xianpeng Hu and Dehua Wang,
*Global solution to the three-dimensional incompressible flow of liquid crystals*, Comm. Math. Phys.**296**(2010), no. 3, 861–880. MR**2628824**, DOI 10.1007/s00220-010-1017-8 - Tao Huang, Changyou Wang, and Huanyao Wen,
*Strong solutions of the compressible nematic liquid crystal flow*, J. Differential Equations**252**(2012), no. 3, 2222–2265. MR**2860617**, DOI 10.1016/j.jde.2011.07.036 - Tao Huang, Changyou Wang, and Huanyao Wen,
*Blow up criterion for compressible nematic liquid crystal flows in dimension three*, Arch. Ration. Mech. Anal.**204**(2012), no. 1, 285–311. MR**2898742**, DOI 10.1007/s00205-011-0476-1 - Fei Jiang and Zhong Tan,
*Global weak solution to the flow of liquid crystals system*, Math. Methods Appl. Sci.**32**(2009), no. 17, 2243–2266. MR**2561116**, DOI 10.1002/mma.1132 - O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,
*Linear and quasilinear equations of parabolic type*, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR**0241822**, DOI 10.1090/mmono/023 - F. M. Leslie,
*Some constitutive equations for liquid crystals*, Arch. Rational Mech. Anal.**28**(1968), no. 4, 265–283. MR**1553506**, DOI 10.1007/BF00251810 - Xiaoli Li and Dehua Wang,
*Global solution to the incompressible flow of liquid crystals*, J. Differential Equations**252**(2012), no. 1, 745–767. MR**2852225**, DOI 10.1016/j.jde.2011.08.045 - Fang-Hua Lin,
*Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena*, Comm. Pure Appl. Math.**42**(1989), no. 6, 789–814. MR**1003435**, DOI 10.1002/cpa.3160420605 - Fang-Hua Lin and Chun Liu,
*Nonparabolic dissipative systems modeling the flow of liquid crystals*, Comm. Pure Appl. Math.**48**(1995), no. 5, 501–537. MR**1329830**, DOI 10.1002/cpa.3160480503 - Fang-Hua Lin and Chun Liu,
*Partial regularity of the dynamic system modeling the flow of liquid crystals*, Discrete Contin. Dynam. Systems**2**(1996), no. 1, 1–22. MR**1367385**, DOI 10.3934/dcds.2011.31.1 - Fang-Hua Lin and Chun Liu,
*Existence of solutions for the Ericksen-Leslie system*, Arch. Ration. Mech. Anal.**154**(2000), no. 2, 135–156. MR**1784963**, DOI 10.1007/s002050000102 - Fanghua Lin and Chun Liu,
*Static and dynamic theories of liquid crystals*, J. Partial Differential Equations**14**(2001), no. 4, 289–330. MR**1883167** - Fanghua Lin, Junyu Lin, and Changyou Wang,
*Liquid crystal flows in two dimensions*, Arch. Ration. Mech. Anal.**197**(2010), no. 1, 297–336. MR**2646822**, DOI 10.1007/s00205-009-0278-x - Pierre-Louis Lions,
*Mathematical topics in fluid mechanics. Vol. 1*, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. MR**1422251** - Chun Liu and Noel J. Walkington,
*Approximation of liquid crystal flows*, SIAM J. Numer. Anal.**37**(2000), no. 3, 725–741. MR**1740379**, DOI 10.1137/S0036142997327282 - Lan Ming Liu and Xian Gao Liu,
*A blow-up criterion for strong solutions to the compressible liquid crystals system*, Chinese Ann. Math. Ser. A**32**(2011), no. 4, 393–406 (Chinese, with English and Chinese summaries). MR**2884827** - X. Liu, L. Liu, and Y. Hao,
*Existence of strong solutions for the compressible Ericksen-Leslie model*. Preprint, 2011. - Xian-Gao Liu and Jie Qing,
*Globally weak solutions to the flow of compressible liquid crystals system*, Discrete Contin. Dyn. Syst.**33**(2013), no. 2, 757–788. MR**2975133**, DOI 10.3934/dcds.2013.33.757 - Yuming Qin and Lan Huang,
*Global existence and regularity of a $1D$ liquid crystal system*, Nonlinear Anal. Real World Appl.**15**(2014), 172–186. MR**3110563**, DOI 10.1016/j.nonrwa.2013.07.003 - Huan Sun and Chun Liu,
*On energetic variational approaches in modeling the nematic liquid crystal flows*, Discrete Contin. Dyn. Syst.**23**(2009), no. 1-2, 455–475. MR**2449088**, DOI 10.3934/dcds.2009.23.455 - Dehua Wang and Cheng Yu,
*Global weak solution and large-time behavior for the compressible flow of liquid crystals*, Arch. Ration. Mech. Anal.**204**(2012), no. 3, 881–915. MR**2917124**, DOI 10.1007/s00205-011-0488-x

## Additional Information

**Xiaoli Li**- Affiliation: Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of China
- Address at time of publication: College of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China
- Email: xlli@bupt.edu.cn
**Dehua Wang**- Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- Email: dwang@math.pitt.edu
- Received by editor(s): January 3, 2012
- Received by editor(s) in revised form: June 27, 2012
- Published electronically: November 12, 2014
- Additional Notes: The first author’s research was supported in part by the National Natural Science Foundation of China under grant 11401036, by the National Natural Science Foundation of China under grants 11271052 and 11471050, by the China Postdoctoral Science Foundation Funded Project under grant 2013T60085, and by the Fundamental Research for the Central Universities No. 2014 RC 0901

The second author’s research was supported in part by the National Science Foundation under grant DMS-0906160, and by the Office of Naval Research under grant N00014-07-1-0668. - © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**367**(2015), 2301-2338 - MSC (2010): Primary 35A05, 76A10, 76D03
- DOI: https://doi.org/10.1090/S0002-9947-2014-05924-2
- MathSciNet review: 3301866