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A generalization of the Babbage functional equation
Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions
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 |
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 [
References:
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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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