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On the inelastic Boltzmann equation for soft potentials with diffusion
On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum
1. | College of Mathematics, Changchun Normal University, Changchun 130032, China |
2. | School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
This paper deals with the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible nematic liquid crystal flows. The global well-posedness of strong solutions with large velocity is established provided that $ \|\rho_0\|_{L^\infty}+\|\nabla d_0\|_{L^3} $ is suitably small. In particular, the initial density may contain vacuum states and even have compact support. Furthermore, the large time behavior of the solution is also obtained.
References:
[1] |
S. Ding, J. Huang and F. Xia,
Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum, Filomat, 27 (2013), 1247-1257.
doi: 10.2298/FIL1307247D. |
[2] |
J. Ericksen,
Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378.
doi: 10.1007/BF00253358. |
[3] |
M. Hong,
Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differ. Equ., 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
[4] |
M. Hong and Y. Mei, Well-posedness of the Ericksen-Leslie system with the Oseen-Frank energy in $L^3_uloc(\mathbb{R}^3)$, Calc. Var. Partial Differ. Equ., 58 (2019), .
doi: 10.1007/s00526-018-1453-4. |
[5] |
M. Hong and Z. Xin,
Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\Bbb R^2$, Adv. Math., 231 (2012), 1364-1400.
doi: 10.1016/j.aim.2012.06.009. |
[6] |
F. Leslie,
Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[7] |
J. Li,
Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94.
doi: 10.1016/j.na.2013.12.023. |
[8] |
J. Li,
Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow, Methods Appl. Anal., 22 (2015), 201-220.
doi: 10.4310/MAA.2015.v22.n2.a4. |
[9] |
J. Li,
Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differ. Equ., 263 (2017), 6512-6536.
doi: 10.1016/j.jde.2017.07.021. |
[10] |
X. Li,
Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two, Discrete Contin. Dyn. Syst., 37 (2017), 4907-4922.
doi: 10.3934/dcds.2017211. |
[11] |
L. Li, Q. Liu and and X. Zhong,
Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum, Nonlinearity, 30 (2017), 4062-4088.
doi: 10.1088/1361-6544/aa8426. |
[12] |
F. Lin, J. Lin and and C. Wang,
Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[13] |
F. Lin and C. Wang,
On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.
doi: 10.1007/s11401-010-0612-5. |
[14] |
F. Lin and C. Wang,
Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, Commun. Pure Appl. Math., 69 (2016), 1532-1571.
doi: 10.1002/cpa.21583. |
[15] |
P. L. Lions, Mathematical topics in fluid mechanics, Vol. I: incompressible models, Oxford University Press, Oxford, 1996.
![]() ![]() |
[16] |
Q. Liu, S. Liu, W. Tan and X. Zhong,
Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differ. Equ., 261 (2016), 6521-6569.
doi: 10.1016/j.jde.2016.08.044. |
[17] |
Q. Liu, C. Wang, X. Zhang and J. Zhou, On optimal boundary control of Ericksen-Leslie system in dimension two, Calc. Var. Partial Differ. Equ, 59 (2020), 64pp.
doi: 10.1007/s00526-019-1676-z. |
[18] |
S. Liu and J. Zhang,
Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2631-2648.
doi: 10.3934/dcdsb.2016065. |
[19] |
B. Lü, X. Shi and and X. Zhong,
Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.
doi: 10.1088/1361-6544/aab31f. |
[20] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, 13 (1959), 115-162.
|
[21] |
R. Temam, Navier-Stokes equations: theory and numerical analysis, Chelsea Publishing, Providence, RI, 2001. Reprint of the 1984 edition.
doi: 10.1090/chel/343. |
[22] |
H. Wen and S. Ding,
Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531.
doi: 10.1016/j.nonrwa.2010.10.010. |
[23] |
H. Yu and P. Zhang,
Global regularity to the 3D incompressible nematic liquid crystal flows with vacuum, Nonlinear Anal., 174 (2018), 209-222.
doi: 10.1016/j.na.2018.04.022. |
[24] |
X. Zhong,
A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum, C. R. Math. Acad. Sci. Paris, 356 (2018), 503-508.
doi: 10.1016/j.crma.2018.04.011. |
show all references
References:
[1] |
S. Ding, J. Huang and F. Xia,
Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum, Filomat, 27 (2013), 1247-1257.
doi: 10.2298/FIL1307247D. |
[2] |
J. Ericksen,
Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378.
doi: 10.1007/BF00253358. |
[3] |
M. Hong,
Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differ. Equ., 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
[4] |
M. Hong and Y. Mei, Well-posedness of the Ericksen-Leslie system with the Oseen-Frank energy in $L^3_uloc(\mathbb{R}^3)$, Calc. Var. Partial Differ. Equ., 58 (2019), .
doi: 10.1007/s00526-018-1453-4. |
[5] |
M. Hong and Z. Xin,
Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\Bbb R^2$, Adv. Math., 231 (2012), 1364-1400.
doi: 10.1016/j.aim.2012.06.009. |
[6] |
F. Leslie,
Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[7] |
J. Li,
Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94.
doi: 10.1016/j.na.2013.12.023. |
[8] |
J. Li,
Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow, Methods Appl. Anal., 22 (2015), 201-220.
doi: 10.4310/MAA.2015.v22.n2.a4. |
[9] |
J. Li,
Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differ. Equ., 263 (2017), 6512-6536.
doi: 10.1016/j.jde.2017.07.021. |
[10] |
X. Li,
Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two, Discrete Contin. Dyn. Syst., 37 (2017), 4907-4922.
doi: 10.3934/dcds.2017211. |
[11] |
L. Li, Q. Liu and and X. Zhong,
Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum, Nonlinearity, 30 (2017), 4062-4088.
doi: 10.1088/1361-6544/aa8426. |
[12] |
F. Lin, J. Lin and and C. Wang,
Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[13] |
F. Lin and C. Wang,
On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.
doi: 10.1007/s11401-010-0612-5. |
[14] |
F. Lin and C. Wang,
Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, Commun. Pure Appl. Math., 69 (2016), 1532-1571.
doi: 10.1002/cpa.21583. |
[15] |
P. L. Lions, Mathematical topics in fluid mechanics, Vol. I: incompressible models, Oxford University Press, Oxford, 1996.
![]() ![]() |
[16] |
Q. Liu, S. Liu, W. Tan and X. Zhong,
Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differ. Equ., 261 (2016), 6521-6569.
doi: 10.1016/j.jde.2016.08.044. |
[17] |
Q. Liu, C. Wang, X. Zhang and J. Zhou, On optimal boundary control of Ericksen-Leslie system in dimension two, Calc. Var. Partial Differ. Equ, 59 (2020), 64pp.
doi: 10.1007/s00526-019-1676-z. |
[18] |
S. Liu and J. Zhang,
Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2631-2648.
doi: 10.3934/dcdsb.2016065. |
[19] |
B. Lü, X. Shi and and X. Zhong,
Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.
doi: 10.1088/1361-6544/aab31f. |
[20] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, 13 (1959), 115-162.
|
[21] |
R. Temam, Navier-Stokes equations: theory and numerical analysis, Chelsea Publishing, Providence, RI, 2001. Reprint of the 1984 edition.
doi: 10.1090/chel/343. |
[22] |
H. Wen and S. Ding,
Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531.
doi: 10.1016/j.nonrwa.2010.10.010. |
[23] |
H. Yu and P. Zhang,
Global regularity to the 3D incompressible nematic liquid crystal flows with vacuum, Nonlinear Anal., 174 (2018), 209-222.
doi: 10.1016/j.na.2018.04.022. |
[24] |
X. Zhong,
A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum, C. R. Math. Acad. Sci. Paris, 356 (2018), 503-508.
doi: 10.1016/j.crma.2018.04.011. |
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