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

doi:10.1007/s10473-024-0305-x

Abstract

CLC Number:

76N10

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.

Trendmd

[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

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

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Xiaoshan Wang, Zhongqian Wang, Zhe Jia. GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH $p$-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE [J]. Acta mathematica scientia,Series B, 2024, 44(3): 909-924.
[2]	Fucai Li, Yue Li, Baoyan Sun. GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^* [J]. Acta mathematica scientia,Series B, 2023, 43(5): 2089-2107.
[3]	Jingpeng, Wu, Xianwen Zhang. THE ENERGY CONSERVATION OF VLASOV-POISSON SYSTEMS^* [J]. Acta mathematica scientia,Series B, 2023, 43(2): 668-674.
[4]	Quansen Jiu, Lin Ma. GLOBAL WEAK SOLUTIONS OF COMPRESSIBLE NAVIER-STOKES-LANDAU-LIFSHITZ-MAXWELL EQUATIONS FOR QUANTUM FLUIDS IN DIMENSION THREE^* [J]. Acta mathematica scientia,Series B, 2023, 43(1): 25-42.
[5]	Yiming HE, Ruizhao ZI. ENERGY CONSERVATION FOR SOLUTIONS OF INCOMPRESSIBLE VISCOELASTIC FLUIDS [J]. Acta mathematica scientia,Series B, 2021, 41(4): 1287-1301.
[6]	Yabo REN, Boling GUO, Shu WANG. GLOBAL WEAK SOLUTIONS TO THE α-MODEL REGULARIZATION FOR 3D COMPRESSIBLE EULER-POISSON EQUATIONS [J]. Acta mathematica scientia,Series B, 2021, 41(3): 679-702.
[7]	Qingyou SUN, Yunguang LU, Christian KLINGENBERG. GLOBAL WEAK SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM [J]. Acta mathematica scientia,Series B, 2020, 40(5): 1185-1194.
[8]	Xueke PU, Boling GUO. INITIAL BOUNDARY VALUE PROBLEM FOR THE 3D MAGNETIC-CURVATURE-DRIVEN RAYLEIGH-TAYLOR MODEL [J]. Acta mathematica scientia,Series B, 2020, 40(2): 529-542.
[9]	Shanshan GUO, Zhong TAN. ON INTEGRABILITY UP TO THE BOUNDARY OF THE WEAK SOLUTIONS TO A NON-NEWTONIAN FLUID [J]. Acta mathematica scientia,Series B, 2019, 39(2): 420-428.
[10]	Jinkai LI, Zhouping XIN. GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D [J]. Acta mathematica scientia,Series B, 2016, 36(4): 973-1014.
[11]	Weiwei WANG. ON GLOBAL BEHAVIOR OF WEAK SOLUTIONS OF COMPRESSIBLE FLOWS OF NEMATIC LIQUID CRYSTALS [J]. Acta mathematica scientia,Series B, 2015, 35(3): 650-672.
[12]	Wei LI, Xing WANG, Nanjing HUANG. DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES [J]. Acta mathematica scientia,Series B, 2015, 35(2): 407-422.
[13]	CHEN Qing, TAN Zhong, WU Guo-Chun. LPS´S CRITERION FOR INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS [J]. Acta mathematica scientia,Series B, 2014, 34(4): 1072-1080.
[14]	Helmer A. FRIIS, Steinar EVJE. WELL-POSEDNESS OF A COMPRESSIBLE GAS-LIQUID MODEL FOR DEEPWATER OIL WELL OPERATIONS [J]. Acta mathematica scientia,Series B, 2014, 34(1): 107-124.
[15]	GAO Li-Ping. STABILITY AND SUPER CONVERGENCE ANALYSIS OF ADI-FDTD FOR THE 2D MAXWELL EQUATIONS IN A LOSSY MEDIUM [J]. Acta mathematica scientia,Series B, 2012, 32(6): 2341-2368.