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Lifting the regionally proximal relation and characterizations of distal extensions
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 |
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.
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. Brzeźniak, E. 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. Brzeźniak, E. 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. Brzeźniak, U. 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. Huang, F. 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. Lin, J. 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. Brzeźniak, E. 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. Brzeźniak, E. 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. Brzeźniak, U. 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. Huang, F. 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. Lin, J. 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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