Large-time behavior of solutions to a \(1D\) liquid crystal system. (English) Zbl 1382.35046
Summary: In this paper, we prove the large-time behavior, as time tends to infinity, of solutions in \(H^i\times H_0^i\times H^{i+1}(i=1,2)\) and \(H^4\times H_0^4 \times H^4\) for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier-Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 35D35 | Strong solutions to PDEs |
| 76A15 | Liquid crystals |
| 35Q30 | Navier-Stokes equations |
References:
| [1] | EricksenJ. Continuum theory of nematic liquid crystals. Res Mechanica. 1961;21:381‐392. |
| [2] | LeslieF. Theory of flow phenomena in liquid crystals adanvances in liquid crystals. Adv Liq Cryst. 1979;4:1‐81. |
| [3] | EricksenJ. Conservation laws for liquid crystals. Trans Soc Rheol.1961;5:22‐34. |
| [4] | LesileF. Some sonstitutive equations for liquid crystals. Arch Rational Mech Anal.1968;28:265‐283. · Zbl 0159.57101 |
| [5] | LinF, LiuC. Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm Pue Appl Math XLV, III. 1995:1-37. |
| [6] | LiuF, LiuC. Partial regularity of the dynamical system modeling the flow of liquid crystals. Discrete Conti Dyna Syst.1996;2:1‐25. |
| [7] | LiuF, LiuC. Existence of solutions for the Ericksen‐Lesile system. Arch Rational Mech Anal.2000;154:135-156. · Zbl 0963.35158 |
| [8] | LiuC, ShenJ. On liquid crystal flows with free‐slip boundary conditions. Discrete Conti Dyna Syst.2001;7:307‐318. · Zbl 1014.76005 |
| [9] | BlancaE, FranciscoG, MarkoR. Reproductivity for a nematic liquid crystal model. Z angew Math Phys. 2006;984-988. · Zbl 1106.35058 |
| [10] | LiuC, WalkingtonN. Approximation of liquid crystal flows. SIAM J Appl Math.2000;37:725‐741. · Zbl 1040.76036 |
| [11] | LiuC, WalkingtonN. Mixed methods for the approximation of liquid crystal flows. Math Modeling and Numer Anal.2002;36:205‐222. · Zbl 1032.76035 |
| [12] | CaldererMC, GolovatyD, LinF, LiuC. Time evolution of nematic liquid crystals with variable degree of orientation. SIAM J Math Anal.2002;33:1033‐1047. · Zbl 1003.35108 |
| [13] | LiuC. Dynamic theory for impressible smectic‐A liquid crystals: existence and regularity. Discrete Conti Dyna Syst.2000;6:591‐608. · Zbl 1021.35083 |
| [14] | DingS. Global solutions of one‐dimensional liquid crystals system. In: A presentation in a Conference at Wuhan University on 1st April; 2009. |
| [15] | QinY, HuangL. Global existence and regularity of a 1D liquid crystal system. Nonlinear Anal. 2014;15:172‐186. · Zbl 1296.35147 |
| [16] | AntonsevSN, KazhikovAV, MonakhovVN. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. North‐Holland: Amsterdam, 1990. · Zbl 0696.76001 |
| [17] | KazhikovAV, ShelukhinVV. Unique global solution with respect to time of initial boundary value problems for one‐dimensional equations of a viscous gas. J Appl Math Mech.1977;41:273‐282. · Zbl 0393.76043 |
| [18] | QinY. Systems, Nonlinear Parabolic‐Hyperbolic Coupled Attractors, Their, Operator Theory Advances in PDEs, vol. 184. Basel‐Boston‐Berlin: Birkhäuser, 2008. · Zbl 1173.35004 |
| [19] | QinY, YuX. Global existence and asymptotic behavior for the compressible Navier‐Stokes equations with a non‐autonomous external force and heat source. Math Meth Appl Sci.2009;32:1011‐1040. · Zbl 1171.35098 |
| [20] | ShenW, ZhengS. On the coupled Cahn‐Hilliard equations. Comm Partial Diff Eqns.1993;18:701‐727. · Zbl 0815.35041 |
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