Weak solutions of non-isothermal nematic liquid crystal flow in dimension three. (English) Zbl 1442.35334
Summary: For any smooth domain \(\Omega \subset \mathbb{R}^3\), we establish the existence of a global weak solution \((\mathbf{u d}, \theta)\) to the simplified, non-isothermal Ericksen-Leslie system modeling the hydrodynamic motion of nematic liquid crystals with variable temperature for any initial and boundary data \((\mathbf{u}_0, \mathbf{d}_0, \theta_0) \in \mathbf{H} \times H^1 (\Omega, \mathbb{S}^2) \times L^1 (\Omega)\), with \(\mathbf{d}_0 (\Omega) \subset \mathbb{S}_+^2\) (the upper half sphere) and \(\text{ess inf}_\Omega \theta_0 > 0\).
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76A10 | Viscoelastic fluids |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 35D30 | Weak solutions to PDEs |
| 35Q56 | Ginzburg-Landau equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
non-isothermal nematic liquid crystals; Ginzburg-Landau approximation; entropy inequalitiesReferences:
| [1] | Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure Appl. Math., 35, 771-831 (1982) · Zbl 0509.35067 · doi:10.1002/cpa.3160350604 |
| [2] | De Anna, F.; Liu, C., Non-isothermal general Ericksen-Leslie system: derivation, analysis and thermodynamic consistency, Arch. Ration. Mech. Anal., 231, 637-717 (2019) · Zbl 1429.76032 · doi:10.1007/s00205-018-1287-4 |
| [3] | Doi, M.; Edwards, S., The Theory of Polymer Dynamics (1986), Oxford: Oxford University Press, Oxford |
| [4] | Ericksen, J., Conservation laws for liquid crystals, Trans. Soc. Rheol., 5, 22-34 (1961) |
| [5] | Ericksen, J., Continuum theory of nematic liquid crystals, Res. Mech., 21, 381-392 (1987) |
| [6] | Evans, L., Partial Differential Equations. Graduate Studies in Mathematics (1998), Providence: American Mathematical Society, Providence · Zbl 0902.35002 |
| [7] | Feireisl, E.; Frémond, M.; Rocca, E.; Shimperna, G., A new approch to nonisothermal models for nematic liquid crystals, Arch. Rational. Mech. Anal., 205, 651-672 (2012) · Zbl 1282.76059 · doi:10.1007/s00205-012-0517-4 |
| [8] | De Gennes, P.; Prost, J., The Physics of Liquid Crystals (1995), Oxford: Oxford University Press, Oxford |
| [9] | Hieber, M.; Prüss, J., Heat kernels and maximal \(L^p-L^q\) estimates for parabolic evolution equations, Commun. Partial Differ. Equ., 22, 1647-1669 (1997) · Zbl 0886.35030 · doi:10.1080/03605309708821314 |
| [10] | Huang, J.; Lin, F.; Wang, C., Regularity and existence of global solutions to the Ericksen-Leslie system in \({\mathbb{R}}^2\), Commun. Math. Phys., 331, 805-850 (2014) · Zbl 1298.35147 · doi:10.1007/s00220-014-2079-9 |
| [11] | Ladyzhenskaya, O.; Solonnikov, VA; Uralćeva, NN, Linear and Quasi-linear Equations of Parabolic Type (1968), Providence: American Mathematical Society, Providence · Zbl 0174.15403 |
| [12] | Leray, J., Sur le mouvement dún liquide visqueux emplissant léspace, Acta. Math., 63, 183-248 (1934) · JFM 60.0726.05 · doi:10.1007/BF02547354 |
| [13] | Leslie, FM, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28, 265-283 (1968) · Zbl 0159.57101 · doi:10.1007/BF00251810 |
| [14] | Li, J.; Xin, Z., Global weak solutions to non-isothermal nematic liquid crystal in 2D, Acta Math. Sci., 36, 973-1014 (2016) · Zbl 1363.35077 · doi:10.1016/S0252-9602(16)30054-6 |
| [15] | Lin, F., Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena, Commun. Pure Appl. Math., 42, 789-814 (1989) · Zbl 0703.35173 · doi:10.1002/cpa.3160420605 |
| [16] | Lin, F., A new proof of the Caffarelli-Kohn-Nirenberg theorem, Commun. Pure Appl. Math., 51, 241-257 (1998) · Zbl 0958.35102 · doi:10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A |
| [17] | Lin, F.; Liu, C., Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48, 501-537 (1995) · Zbl 0842.35084 · doi:10.1002/cpa.3160480503 |
| [18] | Lin, F.; Liu, C., Partial regularity of the dynamic system modeling the flow of liquid crystals, Discret. Contin. Dyn. Syst., 2, 1-22 (1996) · Zbl 0948.35098 · doi:10.3934/dcds.1996.2.1 |
| [19] | Lin, F.; Lin, J.; Wang, C., Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197, 297-336 (2010) · Zbl 1346.76011 · doi:10.1007/s00205-009-0278-x |
| [20] | Lin, F.; Wang, C., Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372, 18 (2014) · Zbl 1353.76004 · doi:10.1098/rsta.2013.0361 |
| [21] | Lin, F.; Wang, C., Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Commun. Pure Appl. Math., 69, 1532-1571 (2016) · Zbl 1349.35299 · doi:10.1002/cpa.21583 |
| [22] | Lin, F.; Wang, C., The Analysis of Harmonic Maps and their Heat Flows (2008), Singapore: World Scientific Publishing Co. Pvt. Ltd, Singapore · Zbl 1203.58004 |
| [23] | Simon, J., Compact sets in the space \(L^p(O.T;B)\), Ann. Mat. Pure Appl., 146, 65-96 (1987) · Zbl 0629.46031 · doi:10.1007/BF01762360 |
| [24] | Sonnet, A.; Virga, E., Dissipative Ordered Fluids: Theories for Liquid Crystals (2012), New York: Springer, New York · Zbl 1258.76002 |
| [25] | Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis (2001), Providence: American Mathematical Society, Providence · Zbl 0981.35001 |
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