不可压双曲型液晶的Ericksen-Leslie方程的大Ericksen数的极限的形式分析

Abstract

Abstract:

In this paper, we consider the parameterized incompressible hyperbolic Ericksen-Leslie system of liquid crystals. Formally, letting the parameter vanish, we prove that the limit system admits a local classical solution. Moreover, we formally obtained an estimate on the difference between the parameterized Ericksen-Leslie's incompressible hyperbolic liquid crystal model and the corresponding limit system, which corresponds to a formal energy estimate of the difference of the classical solutions in $L^2$ space.

Key words: Liquid crystals, Ericksen-Leslie system, Classical solution, Large Ericksen number limit

CLC Number:

O175.25

Feng Cheng. A Formal Analysis on the Large Ericksen Number Limit for the Incompressible Hyperbolic Ericksen-Leslie System of Liquid Crystals[J].Acta mathematica scientia,Series A, 2021, 41(1): 126-141.

Trendmd

References 29

1	De Anna F , Zarnescu A . Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals. J Differential Equations, 2018, 264: 1080- 1118 doi: 10.1016/j.jde.2017.09.031
2	Ericksen J L . Conservation laws for liquid crystals. Trans Soc Rheology, 1961, 5: 23- 34 doi: 10.1122/1.548883
3	Ericksen J L . Continuum theory of nematic liquid crystals. Res Mechanica, 1987, 21: 381- 392
4	Ericksen J L . Liquid crystals with variable degree of orientation. Arch Rational Mech Anal, 1990, 113: 97- 120
5	Feireisl E , Rocca E , Schimperna G , Zarnescu A . On a hyperbolic system arising in liquid crystals modeling. J Hyperbolic Differ Equa, 2018, 15: 15- 35 doi: 10.1142/S0219891618500029
6	Feireisl E , Rocca E , Schimperna G , Zarnescu A . Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy. Annali di Mat Pura ed App, 2015, 194 (5): 1269- 1299 doi: 10.1007/s10231-014-0419-1
7	Ladyzhenskaya O A, Solonnikov N A, Uraltseva N N. Linear and Quasilinear Equations of Parabolic Type. Providence, RI: American Mathematical Society, 1968
8	Leslie F . Theory of Flow Phenomena in Liquid Crystals. New York: Academic Press, 1979: 1- 81
9	Leslie F . Some constitutive equations for liquid crystals. Arch Rational Mech Anal, 1968, 28: 265- 283 doi: 10.1007/BF00251810
10	Majda A J , Bertozzi A L . Vorticity and Incompressible Flow. Cambrige: Cambridge University Press, 2002
11	Gennes P G. The Physics of Liquid Crystals. London: Oxford, 1974
12	Guillén-González F , Rodríguez-Bellido M A . Weak solutions for an initial boundary Q-tensor problem related to liquid crystals. Nonlinear Anal, 2015, 112: 84- 104 doi: 10.1016/j.na.2014.09.011
13	Hardt R , Kinderlehrer D , Lin F H . Existence and partial regularity of static liquid crystal configurations. Comm Math Phys, 1986, 105: 547- 570 doi: 10.1007/BF01238933
14	Hong M C , Xin Z . Global existence of solutions of the liquid crystal fow for the Oseen-Frank model in ${\mathbb R}^2$. Adv Math, 2012, 231: 1364- 1400 doi: 10.1016/j.aim.2012.06.009
15	Huang J , Lin F H , Wang C . Regularity and existence of global solutions to the Ericksen-Leslie system in ${\mathbb R}^2$. Comm Math Phys, 2014, 331: 805- 850 doi: 10.1007/s00220-014-2079-9
16	Kato T . Nonstationary flows of viscous and ideal fluids in ${\mathbb R}^3$. J Funct Anal, 1972, 9: 296- 305 doi: 10.1016/0022-1236(72)90003-1
17	Kato T . On classical solutions of the two-dimensional nonstationary Euler equation. Arch Rational Mech Anal, 1967, 25: 188- 200 doi: 10.1007/BF00251588
18	Ionescu A D, Kemig C E. Local and Global Well-posedness of Periodic KP-I Equations//Bourgain J, Kenig C E, Klainerman S. Mathematical Aspects of Nonlinear Dispersive Equations. Princeton: Princeton University Press, 2009: 181-212
19	Jiang N , Luo Y L . On well-posedness of Ericksen-Leslie's hyperbolic incompressible liquid crystal model. SIAM J Math Anal, 2019, 51 (1): 403- 434 doi: 10.1137/18M1167310
20	Jiang N , Luo Y L , Tang S J . On well-posedness of Ericksen-Leslie's parabolic-hyperbolic liquid crystal model in compressible flow. Math Models Methods Appl Sci, 2019, 29: 121- 183 doi: 10.1142/S0218202519500052
21	Li J , Titi E S , Xin Z . On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in ${\mathbb R}^2$. Math Models Methods Appl Sci, 2016, 26: 803- 822 doi: 10.1142/S0218202516500184
22	Lin F H , Liu C . Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm Pure Appl Math, 1995, 48: 501- 537 doi: 10.1002/cpa.3160480503
23	Lin F H , Liu C . Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete Contin Dynam Systems, 1996, 2: 1- 22 doi: 10.3934/dcds.1996.2.1
24	Lin F H , Liu C . Existence of solutions for the Ericksen-Leslie system. Arch Ration Mech Anal, 2000, 154: 135- 156 doi: 10.1007/s002050000102
25	李强. 分数阶扩散的三维液晶方程的整体正则性. 数学物理学报, 2020, 40A (4): 918- 924 doi: 10.3969/j.issn.1003-3998.2020.04.009
	Li Q . Global regularity for the 3D liquid crystals equations with fractional diffusion. Acta Math Sci, 2020, 40A (4): 918- 924 doi: 10.3969/j.issn.1003-3998.2020.04.009
26	Wang W , Zhang P , Zhang Z . Well-posedness of the Ericksen-Leslie system. Arch Ration Mech Anal, 2013, 210: 837- 855 doi: 10.1007/s00205-013-0659-z
27	Wang M , Wang W . Global existence of weak solution for the 2-D Ericksen-Leslie system. Calc Var Partial Differential Equations, 2014, 51: 915- 962 doi: 10.1007/s00526-013-0700-y
28	Wu H , Xu X , Liu C . On the general Ericksen-Leslie system: Parodi's relation, well-posedness and stability. Arch Ration Mech Anal, 2013, 208 (1): 59- 107 doi: 10.1007/s00205-012-0588-2
29	Wu H , Xu X , Zarnescu A . Dynamics and flow effects in the Beris-Edwards system modeling nematic liquid crystals. Arch Ration Mech Anal, 2019, 231 (2): 1217- 1267 doi: 10.1007/s00205-018-1297-2

[1]	Mingtao Chen,Wenhuo Su,Aibin Zang. Local well-Posedness for the Cauchy Problem of 2D Nonhomogeneous Incompressible and Non-Resistive MHD Equations with Vacuum [J]. Acta mathematica scientia,Series A, 2021, 41(1): 100-125.
[2]	Yue Wang,Hongmin Suo,Wei Wei. Classical Solutions for a Kind of New Kirchhoff-Type Problems Without Boundary Constraint [J]. Acta mathematica scientia,Series A, 2020, 40(4): 857-868.
[3]	Lin Jiaoyan. Uniform Estimates in Time for the Solution to a Fluid-Particle Model [J]. Acta mathematica scientia,Series A, 2018, 38(3): 565-570.
[4]	XIE Sheng-Li. Existence Results of Damped Second Order Impulsive Functional Differential Equations with Infinite Delay [J]. Acta mathematica scientia,Series A, 2015, 35(1): 97-109.
[5]	YE Yu-Lin, DOU Chang-Sheng, JIU Quan-Sen. LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY [J]. Acta mathematica scientia,Series A, 2014, 34(3): 851-871.
[6]	GAO Zhen-Sheng, JIANG Fei, WANG Wei-Wei. Semi-Strong Solutions to Hydrodynamic Flow of Liquid Crystals [J]. Acta mathematica scientia,Series A, 2014, 34(2): 367-377.
[7]	Zhang Quande. A Necessary Condition on Exitence of Global Solutions of Nonlinear Wave Eguations [J]. Acta mathematica scientia,Series A, 1997, 17(1): 31-37.

A Formal Analysis on the Large Ericksen Number Limit for the Incompressible Hyperbolic Ericksen-Leslie System of Liquid Crystals

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 29

Related Articles 7

Metrics

Comments

Recommended 10