# American Institute of Mathematical Sciences

September  2017, 37(9): 4907-4922. doi: 10.3934/dcds.2017211

## Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two

 College of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

* Corresponding author: Xiaoli Li

Received  April 2016 Revised  April 2017 Published  June 2017

Fund Project: The author is supported in part by the National Natural Science Foundation of China under grants 11401036,11271052 and 11471050

This paper is devoted to the study of the initial-boundary value problem for density-dependent incompressible nematic liquid crystal flows with vacuum in a bounded smooth domain of $\mathbb{R}^2$. The system consists of the Navier-Stokes equations, describing the evolution of an incompressible viscous fluid, coupled with various kinematic transport equations for the molecular orientations. Assuming the initial data are sufficiently regular and satisfy a natural compatibility condition, the existence and uniqueness are established for the global strong solution if the initial data are small. We make use of a critical Sobolev inequality of logarithmic type to improve the regularity of the solution. Our result relaxes the assumption in all previous work that the initial density needs to be bounded away from zero.

Citation: Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
##### References:
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Brown (Ed.), Advances in Liquid Crystals, Vol. 2, Academic Press, New York, (1976), 233–398. doi: 10.1016/B978-0-12-025002-8.50012-9. Google Scholar [12] J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392. Google Scholar [13] G. P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. Google Scholar [14] M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5. Google Scholar [15] X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Comm. Math. Phys., 296 (2010), 861-880. doi: 10.1007/s00220-010-1017-8. Google Scholar [16] X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous imcompressible MHD system, J. Diff. Equations, 254 (2013), 511-527. doi: 10.1016/j.jde.2012.08.029. Google Scholar [17] 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. Google Scholar [18] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. Google Scholar [19] O. A. Ladyzhenskaya, N. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar [22] 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. Google Scholar [23] F. H. Lin, Existence of solutions for the Ericksen-Leslie system, Arch. Rat. Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102. Google Scholar [24] 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 [25] F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete and Continuous Dynamic Systems, 2 (1996), 1-22. Google Scholar [26] 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. Google Scholar [27] C. Liu, Dynamic theory for incompressible smectic-A liquid crystals, Discrete and Continuous Dynamic Systems, 6 (2000), 591-608. doi: 10.3934/dcds.2000.6.591. Google Scholar [28] C. Liu and N. J. Walkington, Approximation of liquid crystal flow, SIAM J. Numer. Anal., 37 (2000), 725-741. doi: 10.1137/S0036142997327282. Google Scholar [29] X. Liu and Z. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math., 30 (2009), 1-20. Google Scholar [30] M. Paicu, P. Zhang and Z. Zhang, Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density, Comm. Partial Differential Equations, 38 (2013), 1208-1234. doi: 10.1080/03605302.2013.780079. Google Scholar [31] S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Comm. Partial Diffrential Equations, 27 (2002), 1103-1137. doi: 10.1081/PDE-120004895. Google Scholar [32] 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. Google Scholar [33] D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915. doi: 10.1007/s00205-011-0488-x. Google Scholar [34] 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. Google Scholar

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##### References:
 [1] S. A. Antontesv, A. V. Kazhikov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990. Google Scholar [2] H. Brezis and S. Wainger, A note on limiting cases of Sobolev embedding and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154. Google Scholar [3] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. Google Scholar [4] K. C. Chang, W. Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom., 36 (1992), 507-515. doi: 10.4310/jdg/1214448751. Google Scholar [5] H. Y. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201. doi: 10.1081/PDE-120021191. Google Scholar [6] B. Climent-Ezquerra, F. Guillén-González and M. Rojas-Medar, Reproductivity for a nematic liquid crystal model, Z. angew. Math. Phys., 57 (2006), 984-998. doi: 10.1007/s00033-005-0038-1. Google Scholar [7] R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381. doi: 10.1007/s00021-004-0147-1. Google Scholar [8] R. Danchin and P. B. Mucha, Incompressible flows with piecewise constant density, Arch. Rat. Mech. Anal., 207 (2013), 991-1023. doi: 10.1007/s00205-012-0586-4. Google Scholar [9] B. Desjardins, Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations, Differential and Integral Equations, 10 (1997), 577-586. Google Scholar [10] J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34. doi: 10.1122/1.548883. Google Scholar [11] J. Ericksen, Equilibrium theory for liquid crystals, in: G. Brown (Ed.), Advances in Liquid Crystals, Vol. 2, Academic Press, New York, (1976), 233–398. doi: 10.1016/B978-0-12-025002-8.50012-9. Google Scholar [12] J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392. Google Scholar [13] G. P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. Google Scholar [14] M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5. Google Scholar [15] X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Comm. Math. Phys., 296 (2010), 861-880. doi: 10.1007/s00220-010-1017-8. Google Scholar [16] X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous imcompressible MHD system, J. Diff. Equations, 254 (2013), 511-527. doi: 10.1016/j.jde.2012.08.029. Google Scholar [17] 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. Google Scholar [18] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. Google Scholar [19] O. A. Ladyzhenskaya, N. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar [22] 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. Google Scholar [23] F. H. Lin, Existence of solutions for the Ericksen-Leslie system, Arch. Rat. Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102. Google Scholar [24] 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 [25] F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete and Continuous Dynamic Systems, 2 (1996), 1-22. Google Scholar [26] 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. Google Scholar [27] C. Liu, Dynamic theory for incompressible smectic-A liquid crystals, Discrete and Continuous Dynamic Systems, 6 (2000), 591-608. doi: 10.3934/dcds.2000.6.591. Google Scholar [28] C. Liu and N. J. Walkington, Approximation of liquid crystal flow, SIAM J. Numer. Anal., 37 (2000), 725-741. doi: 10.1137/S0036142997327282. Google Scholar [29] X. Liu and Z. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math., 30 (2009), 1-20. Google Scholar [30] M. Paicu, P. Zhang and Z. Zhang, Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density, Comm. Partial Differential Equations, 38 (2013), 1208-1234. doi: 10.1080/03605302.2013.780079. Google Scholar [31] S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Comm. Partial Diffrential Equations, 27 (2002), 1103-1137. doi: 10.1081/PDE-120004895. Google Scholar [32] 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. Google Scholar [33] D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915. doi: 10.1007/s00205-011-0488-x. Google Scholar [34] 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. Google Scholar
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