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 and Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
References:
[1]

S. A. Antontesv, A. V. Kazhikov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990.

[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.

[3]

L. CaffarelliR. 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.

[4]

K. C. ChangW. 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.

[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.

[6]

B. Climent-EzquerraF. 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.

[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.

[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.

[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. 

[10]

J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34.  doi: 10.1122/1.548883.

[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.

[12]

J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392. 

[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.

[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.

[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.

[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.

[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.

[18]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[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.

[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.

[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.

[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.

[23]

F. H. Lin, Existence of solutions for the Ericksen-Leslie system, Arch. Rat. Mech. Anal., 154 (2000), 135-156.  doi: 10.1007/s002050000102.

[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.

[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. 

[26]

F. H. LinJ. 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.

[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.

[28]

C. Liu and N. J. Walkington, Approximation of liquid crystal flow, SIAM J. Numer. Anal., 37 (2000), 725-741.  doi: 10.1137/S0036142997327282.

[29]

X. Liu and Z. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math., 30 (2009), 1-20. 

[30]

M. PaicuP. 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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

S. A. Antontesv, A. V. Kazhikov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990.

[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.

[3]

L. CaffarelliR. 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.

[4]

K. C. ChangW. 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.

[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.

[6]

B. Climent-EzquerraF. 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.

[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.

[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.

[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. 

[10]

J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34.  doi: 10.1122/1.548883.

[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.

[12]

J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392. 

[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.

[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.

[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.

[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.

[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.

[18]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[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.

[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.

[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.

[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.

[23]

F. H. Lin, Existence of solutions for the Ericksen-Leslie system, Arch. Rat. Mech. Anal., 154 (2000), 135-156.  doi: 10.1007/s002050000102.

[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.

[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. 

[26]

F. H. LinJ. 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.

[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.

[28]

C. Liu and N. J. Walkington, Approximation of liquid crystal flow, SIAM J. Numer. Anal., 37 (2000), 725-741.  doi: 10.1137/S0036142997327282.

[29]

X. Liu and Z. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math., 30 (2009), 1-20. 

[30]

M. PaicuP. 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.

[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.

[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.

[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.

[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.

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