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Some results on the penalised nematic liquid crystals driven by multiplicative noise: weak solution and maximum principle. (English) Zbl 1431.76025

Summary: In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly, we show the pathwise uniqueness of the solution in a 2D domain. In contrast to several works in the deterministic setting, we replace the Ginzburg-Landau function \(𝟙_{|{\mathbf{n}}|\le 1}(|{\mathbf{n}}|^2-1){\mathbf{n}}\) by an appropriate polynomial \(f({\mathbf{n}})\) and we give sufficient conditions on the polynomial \(f\) for these two results to hold. Our third result is a maximum principle type theorem. More precisely, if we consider \(f({\mathbf{n}})=𝟙_{|d|\le 1}(|{\mathbf{n}}|^2-1){\mathbf{n}}\) and if the initial condition \({\mathbf{n}}_0\) satisfies \(|{\mathbf{n}}_0|\le 1\), then the solution \({\mathbf{n}}\) also remains in the unit ball.

MSC:

76A15 Liquid crystals
35Q35 PDEs in connection with fluid mechanics
35R60 PDEs with randomness, stochastic partial differential equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
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[1] Albeverio, S., Brzeźniak, Z., Wu, J.-L.: Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients. J. Math. Anal. Appl. 371, 309-322 (2010) · Zbl 1197.60050 · doi:10.1016/j.jmaa.2010.05.039
[2] Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96, 2nd edn. Birkhäuser, Basel (2011) · Zbl 1226.34002 · doi:10.1007/978-3-0348-0087-7
[3] Atkinson, K., Han, W.: Theoretical Numerical Analysis. A Functional Analysis Framework. Volume 39 of Texts in Applied Mathematics, 3rd edn. Springer, Dordrecht (2009) · Zbl 1181.47078
[4] Bensoussan, A.: Stochastic Navier-Stokes equations. Acta Appl. Math. 38, 267-304 (1995) · Zbl 0836.35115 · doi:10.1007/BF00996149
[5] Brzeźniak, Z., Elworthy, K.D.: Stochastic differential equations on Banach manifolds. Methods Funct. Anal. Topol. 6, 43-84 (2000) · Zbl 0965.58028
[6] Brzeźniak, Z., Goldys, B., Jegaraj, T.: Weak solution of a stochastic Landau-Lifshitz-Gilbert equation. Appl. Math. Res. Express 1, 33 (2012) · Zbl 1272.60041
[7] Brzeźniak, Z., Hausenblas, E., Razafimandimby, P.: Stochastic nonparabolic dissipative systems modeling the flow of liquid crystals: strong solution. In: RIMS Symposium on Mathematical Analysis of Incompressible Flow, February 2013. RIMS Kôkyûroku 1875, pp. 41-73 (2014)
[8] Brzeźniak, Z., Millet, A.: On the stochastic Strichartz estimates and the stochastic nonlinear Schrdinger equation on a compact Riemannian manifold. Potential Anal. 41, 269-315 (2014) · Zbl 1304.58019 · doi:10.1007/s11118-013-9369-2
[9] Cavaterra, C., Rocca, E., Wu, H.: Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows. J. Differ. Equ. 255, 24-57 (2013) · Zbl 1282.35087 · doi:10.1016/j.jde.2013.03.009
[10] Chandrasekhar, S.: Liquid Crystals. Cambridge University Press, Cambridge (1992) · Zbl 0127.41403 · doi:10.1017/CBO9780511622496
[11] Climent-Ezquerra, B., Guillén-González, F., Rojas-Medar, M.A.: Reproductivity for a nematic liquid crystal model. Z. Angew. Math. Phys. 57, 984-998 (2006) · Zbl 1106.35058 · doi:10.1007/s00033-005-0038-1
[12] Constantin, P., Foias, C.: Navier-Stokes Equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988) · Zbl 0687.35071
[13] Dai, M., Schonbek, M.: Asymptotic behavior of solutions to the liquid crystal system in \[H^m(R^3)\] Hm(R3). SIAM J. Math. Anal. 46, 3131-3150 (2014) · Zbl 1307.76026 · doi:10.1137/120895342
[14] Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Vol. 152 of Encyclopedia of Mathematics and Its Applications, 2nd edn. Cambridge University Press, Cambridge (2014) · Zbl 1317.60077 · doi:10.1017/CBO9781107295513
[15] de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1993)
[16] Denis, L., Matoussi, A., Stoica, L.: Maximum principle and comparison theorem for quasi-linear stochastic PDE’s. Electron. J. Probab. 14, 500-530 (2009) · Zbl 1190.60050 · doi:10.1214/EJP.v14-629
[17] Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23-34 (1961) · doi:10.1122/1.548883
[18] Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Related Fields 102, 367-391 (1995) · Zbl 0831.60072 · doi:10.1007/BF01192467
[19] Guillén-González, F., Rojas-Medar, M.: Global solution of nematic liquid crystals models. Comptes Rendus Acad. Sci. Paris Série I 335, 1085-1090 (2002) · Zbl 1022.76005 · doi:10.1016/S1631-073X(02)02620-1
[20] Haroske, D.D., Triebel, H.: Distributions, Sobolev Spaces, Elliptic Equations. EMS Textbooks in Mathematics. European Mathematical Society, Zürich (2008) · Zbl 1133.46001
[21] Hong, M.-C.: Global existence of solutions of the simplified Ericksen-Leslie system in dimension two. Calc. Var. 40, 15-36 (2011) · Zbl 1213.35014 · doi:10.1007/s00526-010-0331-5
[22] Hong, M.-C., Li, J., Xin, Z.: Blow-up criteria of strong solutions to the Ericksen-Leslie system in \[{\mathbb{R}}^3\] R3. Commun. Partial Differ. Equ. 39, 1284-1328 (2014) · Zbl 1327.35319 · doi:10.1080/03605302.2013.871026
[23] Hong, M.-C., Xin, Z.: Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in \[{\mathbb{R}}^2\] R2. Adv. Math. 231, 1364-1400 (2012) · Zbl 1260.35113 · doi:10.1016/j.aim.2012.06.009
[24] Horsthemke, W., Lefever, R.: Noise-Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology. Springer Series in Synergetics, vol. 15. Springer, Berlin (1984) · Zbl 0529.60085
[25] Huang, J., Lin, F., Fanghua, Wang, C.: Regularity and existence of global solutions to the Ericksen-Leslie system in \[{\mathbb{R}}^2\] R2. Commun. Math. Phys. 331, 805-850 (2014) · Zbl 1298.35147 · doi:10.1007/s00220-014-2079-9
[26] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. Volume 24 of North-Holland Mathematical Library, 2nd edn. North-Holland Publishing Co., Amsterdam (1989) · Zbl 0684.60040
[27] Kallenberg, O.: Foundations of Modern Probability. Probability and Its Applications (New York). Springer, New York (1997) · Zbl 0892.60001
[28] Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265-283 (1968) · Zbl 0159.57101 · doi:10.1007/BF00251810
[29] Lin, F.-H., Liu, C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. XLVIII, 501-537 (1995) · Zbl 0842.35084 · doi:10.1002/cpa.3160480503
[30] Lin, F.-H., Liu, C.: Existence of solutions for the Ericksen-Leslie system. Arch. Ration. Mech. Anal. 154, 135-156 (2000) · Zbl 0963.35158 · doi:10.1007/s002050000102
[31] Lin, F., Wang, C.: On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. Chin. Ann. Math. Ser. B 31B, 921-938 (2010) · Zbl 1208.35002 · doi:10.1007/s11401-010-0612-5
[32] 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
[33] Lin, F., Lin, J., Wang, C.: Liquid crystals in two dimensions. Arch. Ration. Mech. Anal. 197, 297-336 (2010) · Zbl 1346.76011 · doi:10.1007/s00205-009-0278-x
[34] Lions, J.L.: Quelques méthodes de résolution de probléms aux limites non-linéaires. Dunod, Paris (1969) · Zbl 0189.40603
[35] Métivier, M.: Semimartingales: A Course on Stochastic Processes. De Gruyter Studies in Mathematics, vol. 2, xi + 287 pp. Walter de Gruyter & Co., Berlin (1982) · Zbl 0503.60054
[36] Ondreját, M.: Stochastic nonlinear wave equations in local Sobolev spaces. Electron. J. Probab. 15, 1041-1091 (2010) · Zbl 1225.60109 · doi:10.1214/EJP.v15-789
[37] Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3, 127-167 (1979) · Zbl 0424.60067 · doi:10.1080/17442507908833142
[38] Protter, P.E.: Stochastic Integration and Differential Equations, Volume 21 of Applications of Mathematics. Stochastic Modelling and Applied Probability, 2nd edn. Springer, Berlin (2004) · Zbl 1041.60005
[39] Ruston, A.F.: Fredholm Theory in Banach Spaces. Volume 86 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1986) · Zbl 0596.47007 · doi:10.1017/CBO9780511569180
[40] San Miguel, M.: Nematic liquid crystals in a stochastic magnetic field: spatial correlations. Phys. Rev. A 32, 3811-3813 (1985) · doi:10.1103/PhysRevA.32.3811
[41] Sagués, F., San Miguel, M.: Dynamics of Fréedericksz transition in a fluctuating magnetic field. Phys. Rev. A. 32, 1843-1851 (1985) · doi:10.1103/PhysRevA.32.1843
[42] Shkoller, S.: Well-posedness and global attractors for liquid crystal on Riemannian manifolds. Commun. Partial Differ. Equ. 27, 1103-1137 (2002) · Zbl 1011.35029 · doi:10.1081/PDE-120004895
[43] Simon, J.: Compact sets in the space \[L^p(0, T;B)\] Lp(0,T;B). Ann. Mat. Pura Appl. (4) 146, 65-96 (1987) · Zbl 0629.46031 · doi:10.1007/BF01762360
[44] Simon, J.: Sobolev, Besov and Nikol’skii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. (4) 157, 117-148 (1990) · Zbl 0727.46018 · doi:10.1007/BF01765315
[45] Temam, R.: Navier-Stokes Equations. North-Holland, Amsterdam (1979) · Zbl 0426.35003
[46] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, vol. 18. North-Holland Publishing Co., Amsterdam (1978) · Zbl 0387.46033
[47] Wang, M., Wang, W.: Global existence of weak solution for the 2-D Ericksen-Leslie system. Calc. Var. Partial Differ. Equ. 51, 915-962 (2014) · Zbl 1303.35077 · doi:10.1007/s00526-013-0700-y
[48] Wang, W., Zhang, P., Zhang, Z.: Well-posedness of the Ericksen-Leslie system. Arch. Ration. Mech. Anal. 210, 837-855 (2013) · Zbl 1360.76030 · doi:10.1007/s00205-013-0659-z
[49] Zeidler, E.: Nonlinear Functional Analysis and Its Applications, II/A: Linear Monotone Operators. Springer, New York (1990) · Zbl 0684.47028 · doi:10.1007/978-1-4612-0981-2
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