Abstract
In two dimensions, we study the compressible hydrodynamic flow of liquid crystals with periodic boundary conditions. As shown by Ding et al. (2013), when the parameter λ → ∞, the solutions to the compressible liquid crystal system approximate that of the incompressible one. Furthermore, Ding et al. (2013) proved that the regular incompressible limit solution is global in time with small enough initial data. In this paper, we show that the solution to the compressible liquid crystal flow also exists for all time, provided that λ is sufficiently large and the initial data are almost incompressible.
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Climent-Ezquerra B, Guillén-González F, Rojas-Medar M. Reproductivity for a nematic liquid crystal model. Z Angew Math Phys, 2006, 57: 984–998
Caffarelli L, Kohn R, Nirenberg L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm Pure Appl Math, 1982, 35: 771–831
Casella E, Secchi P, Trebeschi P. Global existence of 2D slightly compressible viscous magneto-fluid motion. Port Math N S, 2002, 59: 67–89
Choe H J, Jin B J. Global existence of solutions to slightly compressible Navier-Stokes equations in two dimension. Preprint
De Gennes P G. The Physics of Liquid Crystals. New York: Oxford University Press, 1974
Ding S J, Huang J R, Wen H Y, et al. Incompressible limit of the compressible hydrodynamic flow. J Funct Anal, 2013, 264: 1711–1756
Ding S J, Lin J Y, Wang C Y, et al. Compressible hydrodynamic flow of liquid crystals in 1D. Discret Contin Dyn Syst, 2012, 32: 539–563
Ding S J, Wang C Y, Wen H Y. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discret Contin Dyn Syst, 2011, 15: 357–371
Ericksen J. Hydrostatic theory of liquid crystal. Arch Ration Mech Anal, 1962, 9: 371–378
Feireisl E, Rocca E, Schimperna G. On a non-isothermal model for nematic liquid crystals. Nonlinearity, 2011, 24: 243–257
Hagstrom T, Lorenz J. All-time existence of smooth solutions to PDEs of mixed type and the invariant subspace of uniform states. Adv Appl Math, 1995, 16: 219–257
Hagstrom T, Lorenz J. All-time existence of classical solutions for slightly compressible flows. SIAM J Math Anal, 1998, 29: 652–672
Hu X P, Wu H. Global solution to the three-dimensional compressible flow of liquid crystals. ArXiv:1206.2850v1, 2012
Huang T, Wang C Y, Wen H Y. Strong solutions of the compressible nematic liquid crystal flow. J Differential Equations, 2012, 252: 2222–2265
Huang T, Wang C Y, Wen H Y. Blow up criterion for compressible nematic liquid crystal flows in dimension three. Arch Ration Mech Anal, 2012, 204: 285–311
Jiang F, Tan Z. Global weak solution to the flow of liquid crystals system. Math Methods Appl Sci, 2009, 30: 2243–2266
Klainerman S, Majda A. Singular limits of quasilinear hyperbolic system with large paramiters and the incompressible limit of compressible fluids. Comm Pure Appl Math, 1981, 34: 481–524
Kreiss H O, Lorenz H. Initial-Boundary Value Problems and The Navier-Stokes Equations. New York: Academic Press, 1989
Lei Z, Zhou Y. Global existence of classical solution for the two-dimensional Oldroyd model via the incompressible limit. SIAM J Math Anal, 2005, 37: 797–814
Leslie F. Some constitutive equations for liquid crystals. Arch Ration Mech Anal, 1968, 28: 265–283
Li J, Xu Z H, Zhang J W. Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows. ArXiv:1204.4966v1, 2012
Li X L, Wang D H. Global strong solution to the density-dependent incompressible flow of liquid crystals. Trans Amer Math Soc, in press
Lin F H. Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena. Comm Pure Appl Math, 1989, 42: 789–814
Lin F H, Liu C. Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm Pure Appl Math, 1995, 48: 501–537
Lin F H, Liu C. Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals. Discret Contin Dyn Syst, 1996, 2: 1–22
Lin F H, Liu C. Existence of solutions for the Ericksen-Leslie system. Arch Ration Mech Anal, 2000, 154: 135–156
Lin F H, Lin J Y, Wang C Y. Liquid crystal flows in two dimensions. Arch Ration Mech Anal, 2010, 197: 297–336
Liu C, Walkington N J. Mixed methods for the approximation of liquid crystal flows. Math Model Numer Anal, 2002, 36: 205–222
Liu X G, Liu L M, Hao Y H. Existence of strong solutions for the compressible Ericksen-Leslie model. ArXiv:1106. 6140v1, 2011
Liu X G, Liu L M, Hao Y H. A blow-up criterion of strong solutions to the compressible liquid crystals system. Chin Ann Math Ser A, 2011, 32: 393–406
Liu X G, Hao Y H. Incompressible limit of a compressible liquid crystals system. ArXiv:1201.5942, 2012
Liu X G, Zhang Z Y. Existence of the flow of liquid crystals system. Chin Ann Math Ser A, 2009, 30: 1–20
Wang D H, Yu C. Incompressible limit for the compressible flow of liquid crystals. ArXiv:1108.4941v1, 2011
Wang D H, Yu C. Global weak solution and large-time behavior for the compressible flow of liquid crystals. Arch Ration Mech Anal, 2012, 204: 881–915.
Wen H Y, Ding S J. Solutions of incompressible hydrodynamic flow of liquid crystals. Nonlinear Anal: Real World Appl, 2011, 12: 1510–1531
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Ding, S., Huang, J. & Lin, J. Global existence for slightly compressible hydrodynamic flow of liquid crystals in two dimensions. Sci. China Math. 56, 2233–2250 (2013). https://doi.org/10.1007/s11425-013-4620-2
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DOI: https://doi.org/10.1007/s11425-013-4620-2