May  2022, 42(5): 2175-2197. doi: 10.3934/dcds.2021187

Global weak solutions to the stochastic Ericksen–Leslie system in dimension two

1. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

2. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

*Corresponding author: Changyou Wang

Received  February 2021 Published  May 2022 Early access  December 2021

Fund Project: Both authors were supported by NSF grants 1764417 and 2101224

We establish the global existence of weak martingale solutions to the simplified stochastic Ericksen–Leslie system modeling the nematic liquid crystal flow driven by Wiener-type noises on the two-dimensional bounded domains. The construction of solutions is based on the convergence of Ginzburg–Landau approximations. To achieve such a convergence, we first utilize the concentration-cancellation method for the Ericksen stress tensor fields based on a Pohozaev type argument, and then the Skorokhod compactness theorem, which is built upon uniform energy estimates.

Citation: Hengrong Du, Changyou Wang. Global weak solutions to the stochastic Ericksen–Leslie system in dimension two. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2175-2197. doi: 10.3934/dcds.2021187
References:
[1]

A. Bensoussan, Stochastic Navier–Stokes equations, Acta Applicandae Mathematica, 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[2]

Z. Brzeźniak, G. Deugoué and P. A. Razafimandimby, On strong solution to the 2D stochastic Ericksen–Leslie system: A Ginzburg–Landau approximation approach, arXiv preprint, arXiv: 2011.00100. (2020).

[3]

Z. Brzeźniak, G. Deugoué and P. A. Razafimandimby, On the 2D Ericksen-Leslie equations with anisotropic energy and external forces, arXiv preprint, arXiv: 2005.07659, (2020).

[4]

Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby, Strong solution to stochastic penalised nematic liquid crystals model driven by multiplicative Gaussian noise, arXiv: 2004.00590, (2020).

[5]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals, Disc. Contin. Dyn. Syst. Ser. B, 24 (2019), 5785-5802.  doi: 10.3934/dcdsb.2019106.

[6]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, Some results on the penalised nematic liquid crystals driven by multiplicative noise: Weak solution and maximum principle, Stochastics and Partial Differential Equations: Analysis and Computations, 7 (2019), 417-475.  doi: 10.1007/s40072-018-0131-z.

[7]

Z. BrzeźniakU. Manna and A. A. Panda, Martingale solutions of nematic liquid crystals driven by pure jump noise in the Marcus canonical form, J. Diff. Equations, 266 (2019), 6204-6283.  doi: 10.1016/j.jde.2018.11.001.

[8]

A. De Bouard, A. Hocquet and A. Prohl, Existence, uniqueness and regularity for the stochastic Ericksen–Leslie equation, Nonlinearity, 34 (2021), 4057–4114, arXiv: 1902.05921, (2019). doi: 10.1088/1361-6544/ac022e.

[9]

R. J. DiPerna and A. Majda, Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow, J. Amer. Math. Soc., 1 (1988), 59-95.  doi: 10.2307/1990967.

[10]

H. Du, T. Huang and C. Wang, Weak compactness of simplified nematic liquid flows in 2D, arXiv: 2006.04210, (2020).

[11]

J. L. Ericksen, Conservation laws for liquid crystals, Transactions of the Society of Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883.

[12]

J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.

[13]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Prob. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[14]

I. Gyöngy and N. Krylov, Existence of strong solutions for itô's stochastic equations via approximations, Prob. Theory Related fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833.

[15]

M.-C. Hong, Global existence of solutions of the simplified Ericksen–Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5.

[16]

M.-C. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in $\mathbb R^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.

[17]

J. HuangF. Lin and C. Wang, Regularity and existence of global solutions to the Ericksen–Leslie system in $\mathbb{R}^2$, Comm. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9.

[18]

J. Kortum, Concentration-cancellation in the ericksen–leslie model, Calc. Var. Partial Differential Equations, 59 (2020), 1-16.  doi: 10.1007/s00526-020-01849-8.

[19]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.

[20]

F. M. Leslie, Continuum theory for nematic liquid crystals, Continuum Mechanics and Thermodynamics, 4 (1992), 167-175.  doi: 10.1007/BF01130288.

[21]

F. LinJ. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x.

[22]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.

[23]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.

[24]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130361.  doi: 10.1098/rsta.2013.0361.

[25]

F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583.

[26]

J. Simon, Sobolev, besov and nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval, Annali di Matematica Pura ed Applicata, 157 (1990), 117-148.  doi: 10.1007/BF01765315.

[27]

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helvetici, 60 (1985), 558-581.  doi: 10.1007/BF02567432.

show all references

References:
[1]

A. Bensoussan, Stochastic Navier–Stokes equations, Acta Applicandae Mathematica, 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[2]

Z. Brzeźniak, G. Deugoué and P. A. Razafimandimby, On strong solution to the 2D stochastic Ericksen–Leslie system: A Ginzburg–Landau approximation approach, arXiv preprint, arXiv: 2011.00100. (2020).

[3]

Z. Brzeźniak, G. Deugoué and P. A. Razafimandimby, On the 2D Ericksen-Leslie equations with anisotropic energy and external forces, arXiv preprint, arXiv: 2005.07659, (2020).

[4]

Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby, Strong solution to stochastic penalised nematic liquid crystals model driven by multiplicative Gaussian noise, arXiv: 2004.00590, (2020).

[5]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals, Disc. Contin. Dyn. Syst. Ser. B, 24 (2019), 5785-5802.  doi: 10.3934/dcdsb.2019106.

[6]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, Some results on the penalised nematic liquid crystals driven by multiplicative noise: Weak solution and maximum principle, Stochastics and Partial Differential Equations: Analysis and Computations, 7 (2019), 417-475.  doi: 10.1007/s40072-018-0131-z.

[7]

Z. BrzeźniakU. Manna and A. A. Panda, Martingale solutions of nematic liquid crystals driven by pure jump noise in the Marcus canonical form, J. Diff. Equations, 266 (2019), 6204-6283.  doi: 10.1016/j.jde.2018.11.001.

[8]

A. De Bouard, A. Hocquet and A. Prohl, Existence, uniqueness and regularity for the stochastic Ericksen–Leslie equation, Nonlinearity, 34 (2021), 4057–4114, arXiv: 1902.05921, (2019). doi: 10.1088/1361-6544/ac022e.

[9]

R. J. DiPerna and A. Majda, Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow, J. Amer. Math. Soc., 1 (1988), 59-95.  doi: 10.2307/1990967.

[10]

H. Du, T. Huang and C. Wang, Weak compactness of simplified nematic liquid flows in 2D, arXiv: 2006.04210, (2020).

[11]

J. L. Ericksen, Conservation laws for liquid crystals, Transactions of the Society of Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883.

[12]

J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.

[13]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Prob. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[14]

I. Gyöngy and N. Krylov, Existence of strong solutions for itô's stochastic equations via approximations, Prob. Theory Related fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833.

[15]

M.-C. Hong, Global existence of solutions of the simplified Ericksen–Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5.

[16]

M.-C. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in $\mathbb R^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.

[17]

J. HuangF. Lin and C. Wang, Regularity and existence of global solutions to the Ericksen–Leslie system in $\mathbb{R}^2$, Comm. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9.

[18]

J. Kortum, Concentration-cancellation in the ericksen–leslie model, Calc. Var. Partial Differential Equations, 59 (2020), 1-16.  doi: 10.1007/s00526-020-01849-8.

[19]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.

[20]

F. M. Leslie, Continuum theory for nematic liquid crystals, Continuum Mechanics and Thermodynamics, 4 (1992), 167-175.  doi: 10.1007/BF01130288.

[21]

F. LinJ. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x.

[22]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.

[23]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.

[24]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130361.  doi: 10.1098/rsta.2013.0361.

[25]

F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583.

[26]

J. Simon, Sobolev, besov and nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval, Annali di Matematica Pura ed Applicata, 157 (1990), 117-148.  doi: 10.1007/BF01765315.

[27]

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helvetici, 60 (1985), 558-581.  doi: 10.1007/BF02567432.

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