ENERGY CONSERVATION FOR THE WEAK SOLUTIONS TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW

  • Zhong Tan ,
  • Xinliang Li ,
  • Hui Yang
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  • 1. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China;
    2. Shenzhen Research Institute of Xiamen University, Shenzhen 518057, China;
    3. School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China;
    4. College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China
Zhong Tan,E-mail:tan85@xmu.edu.cn; Xinliang Li,E-mail:lixinliangmaths@163.com

Received date: 2022-03-28

  Revised date: 2022-09-12

  Online published: 2024-05-21

Supported by

Tan's research was support by the NSFC (12071391, 12231016) and the Guangdong Basic and Applied Basic Research Foundation (2022A1515010860). Li's research was support by the China Postdoctoral Science Foundation (2023M742401).

Abstract

In this paper, we establish some regularity conditions on the density and velocity fields to guarantee the energy conservation of the weak solutions for the three-dimensional compressible nematic liquid crystal flow in the periodic domain.

Cite this article

Zhong Tan , Xinliang Li , Hui Yang . ENERGY CONSERVATION FOR THE WEAK SOLUTIONS TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW[J]. Acta mathematica scientia, Series B, 2024 , 44(3) : 851 -864 . DOI: 10.1007/s10473-024-0305-x

References

[1] Akramov I, Dębiec T, Skipper J, Wiedemann E. Energy conservation for the compressible Euler and Navier-Stokes equations with vacuum. Anal PDE,2020, 13: 789-811
[2] Bardos C, Titi E S. Onsager's conjecture for the incompressible Euler equations in bounded domains. Arch Ration Mech Anal, 2018, 228: 197-207
[3] Beris A, Edwards B.Thermodynamics of Flowing Systems. Oxford: Oxford University Press, 1994
[4] Buckmaster T, De Lellis C, Isett P, Székelyhidi Jr L. Anomalous dissipation for 1/5-Hölder Euler flows. Ann Math, 2015, 182: 127-172
[5] Buckmaster T, De Lellis C, Székelyhidi Jr L. Dissipative Euler flows with Onsager-critical spatial regularity. Comm Pure Appl Math, 2016, 69: 1613-1670
[6] Chen X, Cheng H. Regularity criterion for 3D nematic liquid crystal flows in terms of finite frequency parts in $\dot{B}^{-1}_{\infty,\infty}$. Bound Value Probl, 2021, 2021: Art 23
[7] Chen X, Fan J. A note on regularity criterion for 3D compressible nematic liquid crystal flows. J Inequal Appl, 2012, 2012: Art 59
[8] Chen M, Liang Z, Wang D, Xu R. Energy equality in compressible fluids with physical boundaries. SIAM J Math Anal,2020, 52:1363-1385
[9] Chandrasekhar S. Liquid Crystals.Cambridge: Cambridge University Press, 1992
[10] Cheskidov A, Constantin P, Friedlander S, Shvydkoy R. Energy conservation and Onsager's conjecture for the Euler equations. Nonlinearity, 2008, 21: 1233-1252
[11] Constantin P, Weinan E, Titi E S. Onsager's conjecture on the energy conservation for solutions of Euler's equation. Commun Math Phys, 1994, 165: 207-209
[12] Daneri S, Runa E, Székelyhidi L. Non-uniqueness for the Euler equations up to Onsager's critical exponent. Ann PDE, 2021, 7(1): Art 8
[13] De Gennes P G. The Physics of Liquid Crystals. Oxford: Oxford University Press, 1974
[14] De Lellis C, László Jr. Dissipative continuous Euler flows. Invent Math, 2013, 193: 377-407
[15] Ericksen J L. Hydrostatic theory of liquid crystal. Arch Rational Mech Anal, 1962, 9: 371-378
[16] Eyink G L. Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer. Physica D: Nonlinear Phenomena, 1994, 78(3/4): 222-240
[17] Ericksen J. Conservation laws for liquid crystals. Trans Soc Rheol, 1961, 5: 23-34
[18] Fan J, Li F. Uniform local well-posedness and regularity criterion for the density-dependent incompressible flow of liquid crystals. Commun Math Sci, 2014, 12: 1185-1197
[19] Fan J, Ozawa T.Regularity criterion for the 3D nematic liquid crystal flows. ISRN Math Anal,2012, 2012: Art 935045
[20] Fan J, Ozawa T. Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals. Discrete Contin Dyn Syst, 2009, 25: 859-867
[21] Fan J, Guo B. Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in $\mathbb{R}^3$. Sci China Ser A, 2008, 51: 1787-1797
[22] Feireisl E, Gwiazda P, Świerczewska A, Wiedemann E. Regularity and energy conservation for the compressible Euler equations. Arch Ration Mech Anal, 2017, 223: 1375-1395
[23] Feireisl E, Rocca E, Schimperna G. On a non-isothermal model for nematic liquid crystal. Nonlinearity, 2011, 24: 243-257
[24] Frank F. Liquid crystals. On the theory of liquid crystals. Discuss Faraday Soc, 1958, 25: 19-28
[25] Gala S, Liu Q, Ragusa M. Logarithmically improved regularity criterion for the nematic liquid crystal flows in $\dot{B}^{-1}_{\infty,\infty}$ space. Comput Math Appl, 2013, 65: 1738-1745
[26] Gao J, Tao Q, Yao Z. Strong solutions to the density-dependent incompressible nematic liquid crystal flows. J Differential Equations, 2016, 260: 3691-3748
[27] Gao Z, Tan Z. Blow-up criterion of classical solutions for the incompressible nematic liquid crystal flows. Acta Math Sci, 2017, 37B: 1632-1638
[28] Guo S, Tan Z. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinet Relat Models, 2015, 8: 691-706
[29] Hardt R, Kinderlehrer D.Mathematical Questions of Liquid Crystal Theory. New York: Springer-Verlag, 1987
[30] Huang T, Wang C, Wen H. Blow up criterion for compressible nematic liquid crystal flows in dimension three. Arch Ration Mech Anal, 2012, 204: 285-311
[31] Isett P. A proof of Onsager's conjecture. Ann Math, 2018, 188: 871-963
[32] Jiang F, Jiang S, Wang D. Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions. Arch Rational Mech Anal, 2014, 214: 403-451
[33] Jiang F, Jiang S, Wang D. On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain. J Funct Anal, 2013, 265: 3369-3397
[34] Jiang F, Tan Z. Global weak solution to the flow of liquid crystals system. Math Methods Appl Sci, 2009, 32: 2243-2266
[35] Leslie F M. Some constitutive equations for liquid crystals. Arch Rational Mech Anal, 1968, 28: 265-283
[36] Leslie F. Some constitutive equations for anisotropic fluids. Quarterly Journal of Mechanics & Applied Mathematics, 1966, 3: 357-370
[37] Leslie F. Some constitutive equations for liquid crystals. Arch Rational Mech Anal, 1968, 28: 265-283
[38] Leslie F. An analysis of a flow instability in nematic liquid crystals. Journal of Physics D Applied Physics, 1976, 9: 925-937
[39] Li Q, Yuan B. A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field. AIMS Math, 2022, 7: 4168-4175
[40] Lin F H. Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Comm Pure Appl Math,1989, 42: 789-814
[41] Lions P L.Mathematical Topics in Fluid Mechanics. Vol 2: Compressible Mdels. New York: Oxford University Press, 1998
[42] Liu Q, Zhao J, Cui S. A regularity criterion for the three-dimensional nematic liquid crystal flow in terms of one directional derivative of the velocity. J Math Phys, 2011, 52: 033102
[43] Liu Q, Zhao J. A regularity criterion for the solution of nematic liquid crystal flows in terms of the $\dot{B}^{-1}_{\infty,\infty}$-norm. J Math Anal Appl, 2013, 407: 557-566
[44] Nirenberg L. On elliptic differential equations. Ann Scuola Norm Sup Pisa Cl Sci, 1959, 13: 115-162
[45] Onsager L. Statistical hydrodynamics. Nuovo Cimento, 1949, 6: 279-287
[46] Oseen C. The theory of liquid crystals. Discuss Faraday Soc, 1933, 29: 883-899
[47] Qian C. Remarks on the regularity criterion for the nematic liquid crystal flows in $\mathbb{R}^3$. Appl Math Comput, 2016, 274: 679-689
[48] Qian C. A further note on the regularity criterion for the 3D nematic liquid crystal flows. Appl Math Comput, 2016, 290: 258-266
[49] Serrin J.The initial value problem for the Navier-Stokes equations// Langer R. Nonlinear Problems. Madison: University of Wisconsin Press, 1963: 69-98
[50] Shinbrot M. The energy equation for the Navier-Stokes system. SIAM J Math Anal, 1974, 5: 948-954
[51] Wang D H, Yu C. Global weak solution and large time behavior for the compressible flow of liquid crystals. Arch Rational Mech Anal, 2012, 204: 881-915
[52] Wang T, Zhao X, Chen Y, Zhang M. Energy conservation for the weak solutions to the equations of compressible magnetohydrodynamic flows in three dimensions. J Math Anal Appl, 2019, 480(2): 123373
[53] Wang X, Liu S. Energy conservation for the weak solutions to the 3D compressible magnetohydrodynamic equations of viscous non-resistive fluids in a bounded domain. Nonlinear Anal: Real World Appl, 2021, 62: 103359
[54] Wang Y, Ye Y. Energy conservation for weak solutions to the 3D Navier-Stokes-Cahn-Hilliard system. Appl Math Lett, 2022, 123: 107587
[55] Wang G, Zuo B. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete Contin Dyn Syst Ser B, 2022, 27(2): 1001-1027
[56] Wang Y, Huang X. On center singularity for compressible spherically symmetric nematic liquid crystal flows. J Differential Equations, 2018, 264: 5197-5220
[57] Wei R, Yao Z, Li Y. Regularity criterion for the nematic liquid crystal flows in terms of velocity. Abstr Appl Anal, 2014, 2014: Art 234809
[58] Yu C. Energy conservation for the weak solutions of the compressible Navier-Stokes equations. Arch Rational Mech Anal, 2017, 225: 1073-1087
[59] Zhang Z, Tang T, Liu L. An Osgood type regularity criterion for the liquid crystal flows. NoDEA Nonlinear Differential Equations Appl, 2014, 21: 253-262
[60] Zhang Z, Yang X. A regularity criterion for the 3D density-dependent incompressible flow of liquid crystals with vacuum. Ann Polon Math, 2015, 115: 165-177
[61] Zhou Y, Fan J. A regularity criterion for the nematic liquid crystal flows. J Inequal Appl, 2010, 2010: Art 589697
[62] Zhou Y, Fan J, Nakamura G. Global strong solution to the density-dependent 2-D liquid crystal flows. Abstr Appl Anal, 2013, 2013: Art 947291
[63] Zöcher H. The effect of a magneticfield on the nematic state. Discuss Faraday Soc, 1933, 29: 945-957
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