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February  2021, 41(2): 921-966. doi: 10.3934/dcds.2020304

Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions

Yi-Long Luo 1, and  Yangjun Ma 2,,

1. 

Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong, 999077, China
2. 

School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

* Corresponding author: Yangjun Ma

Received  April 2020 Published  February 2021 Early access  August 2020

Fund Project: This work is supported by the grants from the National Natural Foundation of China under contract No. 11971360

In this paper we study the incompressible limit of the compressible inertial Qian-Sheng model for liquid crystal flow. We first derive the uniform energy estimates on the Mach number $ \epsilon $ for both the compressible system and its differential system with respect to time under uniformly in $ \epsilon $ small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as $ \epsilon \rightarrow 0 $, so that we establish the global classical solution of the incompressible system by compactness arguments. We emphasize that, on global in time existence of the incompressible inertial Qian-Sheng model under small size of initial data, the range of our assumptions on the coefficients are significantly enlarged, comparing to the results of De Anna and Zarnescu's work [6]. Moreover, we also obtain the convergence rates associated with $ L^2 $-norm with well-prepared initial data.

Keywords: Compressible inertial Qian-Sheng model, Incompressible limit, Uniform bounds, Low Mach number limit, Convergence rate.
Mathematics Subject Classification: Primary: 35D35, 76W05, 76E19, 35B40; Secondary: 76A15.
Citation: Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304
References:
[1] R. A. Adams and J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam) 140. Elsevier/Academic Press, Amsterdam, 2003. 
[2]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[3]

Q. Bie, H. Cui, Q. Wang and Z.-A. Yao, Incompressible limit for the compressible flow of liquid crystals in Lp type critical Besov spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2879-2910.  doi: 10.3934/dcds.2018124.

[4]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[5]

Y. Cai and W. Wang, Global well-posedness for the three dimensional simplified inertial Ericksen-Leslie systems near equilibrium, J. Funct. Anal., 279 (2020), 108521, 38 pp. doi: 10.1016/j.jfa.2020.108521.

[6]

F. De Anna and A. Zarnescu, Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals, J. Differential Equations, 264 (2018), 1080-1118.  doi: 10.1016/j.jde.2017.09.031.

[7]

S. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962.

[8]

S. Ding, J. Huang, H. Wen and R. Zi, Incompressible limit of the compressible nematic liquid crystal flow, J. Funct. Anal., 264 (2013), 1711-1756.  doi: 10.1016/j.jfa.2013.01.011.

[9]

E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, On a hyperbolic system arising in liquid crystals modeling, J. Hyperbolic Differ. Equ., 15 (2018), 15-35.  doi: 10.1142/S0219891618500029.

[10]

E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl., 76 (1997), 477-498.  doi: 10.1016/S0021-7824(97)89959-X.

[11]

L. Guo, N. Jiang, F. C. Li, Y.-L. Luo and S. J. Tang, Incompressible limit for the compressible Ericksen-Leslie's parabolic-hyperbolic liquid crystal flow, arXiv: 1911.04315v1.

[12]

J. X. Huang, N. Jiang, Y.-L. Luo and L. F. Zhao, Small data global regularity for 3-D Ericksen-Leslie's hyperbolic liquid crystal model without kinematic transport, arXiv: 1812.08341.

[13]

J. X. Huang, N. Jiang, Y.-L. Luo and L. F. Zhao, Small data global regularity for simplified 3-D Ericksen-Leslie's compressible hyperbolic liquid crystal model, arXiv: 1905.04884.

[14]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys., 297 (2010), 371-400.  doi: 10.1007/s00220-010-0992-0.

[15]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the nonisentropic ideal magnetohydrodynamic equations, SIAM J. Math. Anal., 48 (2016), 302-319.  doi: 10.1137/15M102842X.

[16]

S. Jiang, Q. Ju, F. Li and Z.-P. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[17]

N. Jiang and Y.-L. Luo, On well-posedness of Ericksen-Leslie's hyperbolic incompressible liquid crystal model, SIAM J. Math. Anal., 51 (2019), 403-434.  doi: 10.1137/18M1167310.

[18]

N. Jiang, Y.-L. Luo, Y. J. Ma and S. J. Tang, Entropy inequality and energy dissipation of inertial Qian-Sheng model of liquid crystals, Preprint, 2020.

[19]

N. Jiang, Y.-L. Luo and S. Tang, On well-posedness of Ericksen-Leslie's parabolic-hyperbolic liquid crystal model in compressible flow, Math. Models Methods Appl. Sci., 29 (2019), 121-183.  doi: 10.1142/S0218202519500052.

[20]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[21]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651.  doi: 10.1002/cpa.3160350503.

[22]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.

[23] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Oxford Lecture Series in Mathematics and its Applications 3, The Clarendon Press, Oxford University Press, New York, 1996. 
[24]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.

[25]

Y. J. Ma, Global well-posedness of incompressible non-inertial Qian-Sheng model., Discrete Contin. Dyn. Syst., 40 (2020), 4479-4496.  doi: 10.3934/dcds.2020187.

[26] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. 
[27]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[28]

T. Qian and P. Sheng, Generalized hydrodynamic equations for nematic liquid crystals, Phys. Rev. E, 58 (1998), 7475-7485.  doi: 10.1103/PhysRevE.58.7475.

[29]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75.  doi: 10.1007/BF01210792.

[30]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.  doi: 10.1006/jdeq.1994.1157.

[31]

J. V. Selinger, Introduction to the Theory of Soft Matter: From Ideal gas to Liquid Crystals, Springer International Publishing, Switzerland, 2016.

[32]

J. Simon, Compact sets in the space $L^p(0; T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[33]

S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ., 26 (1986), 323-331.  doi: 10.1215/kjm/1250520925.

[34]

H. Wu, X. Xu and A. Zarnescu, Dynamics and flow effects in the Beris-Edwards system modeling nematic liquid crystals, Arch. Ration. Mech. Anal., 231 (2019), 1217-1267.  doi: 10.1007/s00205-018-1297-2.

[35]

D. H. Wang and C. Yu, Incompressible limit for the compressible flow of liquid crystals, J. Math. Fluid Mech., 16 (2014), 771-786.  doi: 10.1007/s00021-014-0185-2.

[36]

X. Yang, Uniform well-posedness and low Mach number limit to the compressible nematic liquid crystal flows in a bounded domain, Nonlinear Anal., 120 (2015), 118-126.  doi: 10.1016/j.na.2015.03.010.

[37]

L. Zeng, G. Ni and X. Ai, Low Mach number limit of global solutions to 3-D compressible nematic liquid crystal flows with Dirichlet boundary condition, Math. Methods Appl. Sci., 42 (2019), 2053-2068.  doi: 10.1002/mma.5499.

show all references

References:
[1] R. A. Adams and J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam) 140. Elsevier/Academic Press, Amsterdam, 2003. 
[2]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[3]

Q. Bie, H. Cui, Q. Wang and Z.-A. Yao, Incompressible limit for the compressible flow of liquid crystals in Lp type critical Besov spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2879-2910.  doi: 10.3934/dcds.2018124.

[4]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[5]

Y. Cai and W. Wang, Global well-posedness for the three dimensional simplified inertial Ericksen-Leslie systems near equilibrium, J. Funct. Anal., 279 (2020), 108521, 38 pp. doi: 10.1016/j.jfa.2020.108521.

[6]

F. De Anna and A. Zarnescu, Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals, J. Differential Equations, 264 (2018), 1080-1118.  doi: 10.1016/j.jde.2017.09.031.

[7]

S. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962.

[8]

S. Ding, J. Huang, H. Wen and R. Zi, Incompressible limit of the compressible nematic liquid crystal flow, J. Funct. Anal., 264 (2013), 1711-1756.  doi: 10.1016/j.jfa.2013.01.011.

[9]

E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, On a hyperbolic system arising in liquid crystals modeling, J. Hyperbolic Differ. Equ., 15 (2018), 15-35.  doi: 10.1142/S0219891618500029.

[10]

E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl., 76 (1997), 477-498.  doi: 10.1016/S0021-7824(97)89959-X.

[11]

L. Guo, N. Jiang, F. C. Li, Y.-L. Luo and S. J. Tang, Incompressible limit for the compressible Ericksen-Leslie's parabolic-hyperbolic liquid crystal flow, arXiv: 1911.04315v1.

[12]

J. X. Huang, N. Jiang, Y.-L. Luo and L. F. Zhao, Small data global regularity for 3-D Ericksen-Leslie's hyperbolic liquid crystal model without kinematic transport, arXiv: 1812.08341.

[13]

J. X. Huang, N. Jiang, Y.-L. Luo and L. F. Zhao, Small data global regularity for simplified 3-D Ericksen-Leslie's compressible hyperbolic liquid crystal model, arXiv: 1905.04884.

[14]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys., 297 (2010), 371-400.  doi: 10.1007/s00220-010-0992-0.

[15]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the nonisentropic ideal magnetohydrodynamic equations, SIAM J. Math. Anal., 48 (2016), 302-319.  doi: 10.1137/15M102842X.

[16]

S. Jiang, Q. Ju, F. Li and Z.-P. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[17]

N. Jiang and Y.-L. Luo, On well-posedness of Ericksen-Leslie's hyperbolic incompressible liquid crystal model, SIAM J. Math. Anal., 51 (2019), 403-434.  doi: 10.1137/18M1167310.

[18]

N. Jiang, Y.-L. Luo, Y. J. Ma and S. J. Tang, Entropy inequality and energy dissipation of inertial Qian-Sheng model of liquid crystals, Preprint, 2020.

[19]

N. Jiang, Y.-L. Luo and S. Tang, On well-posedness of Ericksen-Leslie's parabolic-hyperbolic liquid crystal model in compressible flow, Math. Models Methods Appl. Sci., 29 (2019), 121-183.  doi: 10.1142/S0218202519500052.

[20]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[21]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651.  doi: 10.1002/cpa.3160350503.

[22]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.

[23] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Oxford Lecture Series in Mathematics and its Applications 3, The Clarendon Press, Oxford University Press, New York, 1996. 
[24]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.

[25]

Y. J. Ma, Global well-posedness of incompressible non-inertial Qian-Sheng model., Discrete Contin. Dyn. Syst., 40 (2020), 4479-4496.  doi: 10.3934/dcds.2020187.

[26] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. 
[27]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[28]

T. Qian and P. Sheng, Generalized hydrodynamic equations for nematic liquid crystals, Phys. Rev. E, 58 (1998), 7475-7485.  doi: 10.1103/PhysRevE.58.7475.

[29]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75.  doi: 10.1007/BF01210792.

[30]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.  doi: 10.1006/jdeq.1994.1157.

[31]

J. V. Selinger, Introduction to the Theory of Soft Matter: From Ideal gas to Liquid Crystals, Springer International Publishing, Switzerland, 2016.

[32]

J. Simon, Compact sets in the space $L^p(0; T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[33]

S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ., 26 (1986), 323-331.  doi: 10.1215/kjm/1250520925.

[34]

H. Wu, X. Xu and A. Zarnescu, Dynamics and flow effects in the Beris-Edwards system modeling nematic liquid crystals, Arch. Ration. Mech. Anal., 231 (2019), 1217-1267.  doi: 10.1007/s00205-018-1297-2.

[35]

D. H. Wang and C. Yu, Incompressible limit for the compressible flow of liquid crystals, J. Math. Fluid Mech., 16 (2014), 771-786.  doi: 10.1007/s00021-014-0185-2.

[36]

X. Yang, Uniform well-posedness and low Mach number limit to the compressible nematic liquid crystal flows in a bounded domain, Nonlinear Anal., 120 (2015), 118-126.  doi: 10.1016/j.na.2015.03.010.

[37]

L. Zeng, G. Ni and X. Ai, Low Mach number limit of global solutions to 3-D compressible nematic liquid crystal flows with Dirichlet boundary condition, Math. Methods Appl. Sci., 42 (2019), 2053-2068.  doi: 10.1002/mma.5499.

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