doi: 10.3934/cpaa.2022093
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Local well-posedness to the 2D Cauchy problem of non-isothermal nonhomogeneous nematic liquid crystal flows with vacuum at infinity

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author

Received  February 2022 Revised  April 2022 Early access May 2022

Fund Project: This research was partially supported by National Natural Science Foundation of China (Nos. 11901474, 12071359) and Exceptional Young Talents Project of Chongqing Talent (No. cstc2021ycjh-bgzxm0153)

This paper concerns the Cauchy problem of non-isothermal nonhomogeneous nematic liquid crystal flows in $ \mathbb{R}^2 $ with zero density at infinity. By spatial weighted energy method and a Hardy type inequality, we show the local existence and uniqueness of strong solutions provided that the initial density and the gradient of orientation decay not too slowly at infinity.

Citation: Hong Chen, Xin Zhong. Local well-posedness to the 2D Cauchy problem of non-isothermal nonhomogeneous nematic liquid crystal flows with vacuum at infinity. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022093
References:
[1]

S. AgmonA. Douglis and and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.

[2]

S. AgmonA. Douglis and and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅱ, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[3]

D. Bian and Y. Xiao, Global solution to the nematic liquid crystal flows with heat effect, J. Differ. Equ., 263 (2017), 5298-5329.  doi: 10.1016/j.jde.2017.06.019.

[4]

D. Bian and Y. Xiao, Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1243-1272.  doi: 10.3934/dcdsb.2020161.

[5]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differ. Equ., 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.

[6]

F. De Anna and C. Liu, Non-isothermal general Ericksen-Leslie system: derivation, analysis and thermodynamic consistency, Arch. Ration. Mech. Anal., 231 (2019), 637-717.  doi: 10.1007/s00205-018-1287-4.

[7]

H. DuY. Li and and C. Wang, Weak solutions of non-isothermal nematic liquid crystal flow in dimension three, J. Elliptic Parabol. Equ., 6 (2020), 71-98.  doi: 10.1007/s41808-020-00055-z.

[8]

E. FeireislM. FrémondE. Rocca and and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Ration. Mech. Anal., 205 (2012), 651-672.  doi: 10.1007/s00205-012-0517-4.

[9]

E. FeireislE. Rocca and and G. Schimperna, On a non-isothermal model for nematic liquid crystals, Nonlinearity, 24 (2011), 243-257.  doi: 10.1088/0951-7715/24/1/012.

[10]

E. FeireislE. RoccaG. Schimperna and and A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Commun. Math. Sci., 12 (2014), 317-343.  doi: 10.4310/CMS.2014.v12.n2.a6.

[11]

E. FeireislG. SchimpernaE. Rocca and and A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy, Ann. Mat. Pura Appl., 194 (2015), 1269-1299.  doi: 10.1007/s10231-014-0419-1.

[12]

M. Hieber and J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress Ⅰ: the incompressible isotropic case, Math. Ann., 369 (2017), 977-996.  doi: 10.1007/s00208-016-1453-7.

[13]

M. Hieber and J. Prüss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flows, Handbook of mathematical analysis in mechanics of viscous fluids, 1075–1134, Springer, Cham, 2018.,

[14]

T. HuangC. Wang and and H. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Differ. Equ., 252 (2012), 2222-2265.  doi: 10.1016/j.jde.2011.07.036.

[15]

O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, R.I., 1968.

[16]

J. Li and Z. Xin, Global existence of weak solutions to the non-isothermal nematic liquid crystals in 2D, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 973-1014.  doi: 10.1016/S0252-9602(16)30054-6.

[17]

Q. Li and C. Wang, Local well-posedness of nonhomogeneous incompressible liquid crystals model without compatibility condition, Nonlinear Anal. Real World Appl., 65 (2022), 103474.  doi: 10.1016/j.nonrwa.2021.103474.

[18]

Z. Liang, Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differ. Equ., 258 (2015), 2633-2654.  doi: 10.1016/j.jde.2014.12.015.

[19]

Z. Liang and J. Shuai, Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equation in two dimensions, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5383-5405.  doi: 10.3934/dcdsb.2020348.

[20]

F. 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.

[21]

F. Lin and C. Wang, The analysis of harmonic maps and their heat flows, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/6679.

[22]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361.  doi: 10.1098/rsta.2013.0361.

[23] P. L. Lions, Mathematical topics in fluid mechanics, vol.Ⅰ: incompressible models, Oxford University Press, Oxford, 1996. 
[24]

Q. LiuS. LiuW. Tan and 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.

[25]

Y. Liu, Global regularity to the 2D non-isothermal inhomogeneous nematic liquid crystal flows, Appl. Anal., (2020), 1–21. doi: 10.1080/00036811.2020.1819534.

[26]

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.

[27]

A. M. Sonnet and E. G. Virga, Dissipative ordered fluids. Theories for liquid crystals, Springer, New York, 2012. doi: 10.1007/978-0-387-87815-7.

[28]

X. Zhao and M. Zhu, Strong solutions to the density-dependent incompressible nematic liquid crystal flows with heat effect, Bull. Malays. Math. Sci. Soc., 44 (2021), 1579-1611.  doi: 10.1007/s40840-020-01026-2.

show all references

References:
[1]

S. AgmonA. Douglis and and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.

[2]

S. AgmonA. Douglis and and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅱ, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[3]

D. Bian and Y. Xiao, Global solution to the nematic liquid crystal flows with heat effect, J. Differ. Equ., 263 (2017), 5298-5329.  doi: 10.1016/j.jde.2017.06.019.

[4]

D. Bian and Y. Xiao, Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1243-1272.  doi: 10.3934/dcdsb.2020161.

[5]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differ. Equ., 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.

[6]

F. De Anna and C. Liu, Non-isothermal general Ericksen-Leslie system: derivation, analysis and thermodynamic consistency, Arch. Ration. Mech. Anal., 231 (2019), 637-717.  doi: 10.1007/s00205-018-1287-4.

[7]

H. DuY. Li and and C. Wang, Weak solutions of non-isothermal nematic liquid crystal flow in dimension three, J. Elliptic Parabol. Equ., 6 (2020), 71-98.  doi: 10.1007/s41808-020-00055-z.

[8]

E. FeireislM. FrémondE. Rocca and and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Ration. Mech. Anal., 205 (2012), 651-672.  doi: 10.1007/s00205-012-0517-4.

[9]

E. FeireislE. Rocca and and G. Schimperna, On a non-isothermal model for nematic liquid crystals, Nonlinearity, 24 (2011), 243-257.  doi: 10.1088/0951-7715/24/1/012.

[10]

E. FeireislE. RoccaG. Schimperna and and A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Commun. Math. Sci., 12 (2014), 317-343.  doi: 10.4310/CMS.2014.v12.n2.a6.

[11]

E. FeireislG. SchimpernaE. Rocca and and A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy, Ann. Mat. Pura Appl., 194 (2015), 1269-1299.  doi: 10.1007/s10231-014-0419-1.

[12]

M. Hieber and J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress Ⅰ: the incompressible isotropic case, Math. Ann., 369 (2017), 977-996.  doi: 10.1007/s00208-016-1453-7.

[13]

M. Hieber and J. Prüss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flows, Handbook of mathematical analysis in mechanics of viscous fluids, 1075–1134, Springer, Cham, 2018.,

[14]

T. HuangC. Wang and and H. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Differ. Equ., 252 (2012), 2222-2265.  doi: 10.1016/j.jde.2011.07.036.

[15]

O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, R.I., 1968.

[16]

J. Li and Z. Xin, Global existence of weak solutions to the non-isothermal nematic liquid crystals in 2D, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 973-1014.  doi: 10.1016/S0252-9602(16)30054-6.

[17]

Q. Li and C. Wang, Local well-posedness of nonhomogeneous incompressible liquid crystals model without compatibility condition, Nonlinear Anal. Real World Appl., 65 (2022), 103474.  doi: 10.1016/j.nonrwa.2021.103474.

[18]

Z. Liang, Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differ. Equ., 258 (2015), 2633-2654.  doi: 10.1016/j.jde.2014.12.015.

[19]

Z. Liang and J. Shuai, Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equation in two dimensions, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5383-5405.  doi: 10.3934/dcdsb.2020348.

[20]

F. 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.

[21]

F. Lin and C. Wang, The analysis of harmonic maps and their heat flows, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/6679.

[22]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361.  doi: 10.1098/rsta.2013.0361.

[23] P. L. Lions, Mathematical topics in fluid mechanics, vol.Ⅰ: incompressible models, Oxford University Press, Oxford, 1996. 
[24]

Q. LiuS. LiuW. Tan and 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.

[25]

Y. Liu, Global regularity to the 2D non-isothermal inhomogeneous nematic liquid crystal flows, Appl. Anal., (2020), 1–21. doi: 10.1080/00036811.2020.1819534.

[26]

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.

[27]

A. M. Sonnet and E. G. Virga, Dissipative ordered fluids. Theories for liquid crystals, Springer, New York, 2012. doi: 10.1007/978-0-387-87815-7.

[28]

X. Zhao and M. Zhu, Strong solutions to the density-dependent incompressible nematic liquid crystal flows with heat effect, Bull. Malays. Math. Sci. Soc., 44 (2021), 1579-1611.  doi: 10.1007/s40840-020-01026-2.

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