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

Yi-Long Luo; Yangjun Ma

doi:10.3934/dcds.2020304

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2021, Volume 41, Issue 2: 921-966. Doi: 10.3934/dcds.2020304

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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
^* Corresponding author: Yangjun Ma

Received: April 2020

Early access: August 2020

Published: February 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35D35, 76W05, 76E19, 35B40; Secondary: 76A15.

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[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.