# American Institute of Mathematical Sciences

December  2016, 9(6): 1913-1937. doi: 10.3934/dcdss.2016078

## Well-posedness for the three-dimensional compressible liquid crystal flows

 1 College of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China 2 Institute of Applied Physics & Computational Math., Beijing 100088

Received  July 2015 Revised  September 2016 Published  November 2016

This paper is concerned with the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$. Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and uniqueness is established for the local strong solution with large initial data and also for the global strong solution with initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.
Citation: Xiaoli Li, Boling Guo. Well-posedness for the three-dimensional compressible liquid crystal flows. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1913-1937. doi: 10.3934/dcdss.2016078
##### References:
 [1] K. C. Chang, W. Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Diff. Geom., 36 (1992), 507-515. [2] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Diff. Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9. [3] Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value prob lems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004. [4] Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Diff. Equations, 228 (2006), 377-411. doi: 10.1016/j.jde.2006.05.001. [5] S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Conti. Dyna. Sys. Ser. B, 15 (2011), 357-371. doi: 10.3934/dcdsb.2011.15.357. [6] S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Conti. Dyna. Sys., 32 (2012), 539-563. doi: 10.3934/dcds.2012.32.539. [7] J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34. doi: 10.1122/1.548883. [8] J. Ericksen, Equilibrium theory for liquid crystals, in: G. Brown (Ed.), Advances in Liquid Crystals, Academic Press, New York, 2 (1976), 233-298. doi: 10.1016/B978-0-12-025002-8.50012-9. [9] J. Ericksen, Continuum theory of nematic liquid crystals, Molecular Crystals, 7 (2007), 153-164. doi: 10.1080/15421406908084869. [10] M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Diff. Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5. [11] X. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Anal., 45 (2013), 2678-2699, arXiv:1206.2850. doi: 10.1137/120898814. [12] T. Huang, C. Wang and H. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Diff. Equations, 252 (2012), 2222-2265. doi: 10.1016/j.jde.2011.07.036. [13] T. Huang, C. Wang and H. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three, Arch. Rational Mech. Anal., 204 (2012), 285-311. doi: 10.1007/s00205-011-0476-1. [14] F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Meth. Appl. Sci., 32 (2009), 2243-2266. doi: 10.1002/mma.1132. [15] F. M. Leslie, Theory of flow phenomena in liquid crystals, in: G. Brown (Ed.), Advances in Liquid Crystals, Academic Press, New York, 4 (1979), 1-81. doi: 10.1016/B978-0-12-025004-2.50008-9. [16] X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals, Trans. Amer. Math. Soc., 367 (2015), 2301-2338. doi: 10.1090/S0002-9947-2014-05924-2. [17] F. H. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [18] F. H. Lin, Existence of solutions for the Ericksen-Leslie system, Arch. Rat. Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102. [19] 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. [20] F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Conti. Dyna. Sys., 2 (1996), 1-22. [21] F. H. 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. [22] C. Liu, Dynamic theory for incompressible smectic-A liquid crystals, Discrete Conti. Dyna. Sys., 6 (2000), 591-608. doi: 10.3934/dcds.2000.6.591. [23] X. Liu, L. Liu and Y. Hao, Existence of strong solutions for the compressible Ericksen-Leslie model,, , (). [24] X. Liu and Z. Zhang, {Existence of the flow of liquid crystals system, Chinese Ann. Math., 30 (2009), 1-20. [25] S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Comm. Partial Diff. Equations, 27 (2001), 1103-1137. doi: 10.1081/PDE-120004895. [26] C. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal., 200 (2011), 1-19. doi: 10.1007/s00205-010-0343-5. [27] H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Analysis: Real World Applications, 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010.

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##### References:
 [1] K. C. Chang, W. Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Diff. Geom., 36 (1992), 507-515. [2] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Diff. Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9. [3] Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value prob lems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004. [4] Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Diff. Equations, 228 (2006), 377-411. doi: 10.1016/j.jde.2006.05.001. [5] S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Conti. Dyna. Sys. Ser. B, 15 (2011), 357-371. doi: 10.3934/dcdsb.2011.15.357. [6] S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Conti. Dyna. Sys., 32 (2012), 539-563. doi: 10.3934/dcds.2012.32.539. [7] J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34. doi: 10.1122/1.548883. [8] J. Ericksen, Equilibrium theory for liquid crystals, in: G. Brown (Ed.), Advances in Liquid Crystals, Academic Press, New York, 2 (1976), 233-298. doi: 10.1016/B978-0-12-025002-8.50012-9. [9] J. Ericksen, Continuum theory of nematic liquid crystals, Molecular Crystals, 7 (2007), 153-164. doi: 10.1080/15421406908084869. [10] M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Diff. Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5. [11] X. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Anal., 45 (2013), 2678-2699, arXiv:1206.2850. doi: 10.1137/120898814. [12] T. Huang, C. Wang and H. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Diff. Equations, 252 (2012), 2222-2265. doi: 10.1016/j.jde.2011.07.036. [13] T. Huang, C. Wang and H. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three, Arch. Rational Mech. Anal., 204 (2012), 285-311. doi: 10.1007/s00205-011-0476-1. [14] F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Meth. Appl. Sci., 32 (2009), 2243-2266. doi: 10.1002/mma.1132. [15] F. M. Leslie, Theory of flow phenomena in liquid crystals, in: G. Brown (Ed.), Advances in Liquid Crystals, Academic Press, New York, 4 (1979), 1-81. doi: 10.1016/B978-0-12-025004-2.50008-9. [16] X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals, Trans. Amer. Math. Soc., 367 (2015), 2301-2338. doi: 10.1090/S0002-9947-2014-05924-2. [17] F. H. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [18] F. H. Lin, Existence of solutions for the Ericksen-Leslie system, Arch. Rat. Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102. [19] 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. [20] F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Conti. Dyna. Sys., 2 (1996), 1-22. [21] F. H. 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. [22] C. Liu, Dynamic theory for incompressible smectic-A liquid crystals, Discrete Conti. Dyna. Sys., 6 (2000), 591-608. doi: 10.3934/dcds.2000.6.591. [23] X. Liu, L. Liu and Y. Hao, Existence of strong solutions for the compressible Ericksen-Leslie model,, , (). [24] X. Liu and Z. Zhang, {Existence of the flow of liquid crystals system, Chinese Ann. Math., 30 (2009), 1-20. [25] S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Comm. Partial Diff. Equations, 27 (2001), 1103-1137. doi: 10.1081/PDE-120004895. [26] C. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal., 200 (2011), 1-19. doi: 10.1007/s00205-010-0343-5. [27] H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Analysis: Real World Applications, 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010.
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