# American Institute of Mathematical Sciences

August  2022, 15(8): 2117-2133. doi: 10.3934/dcdss.2022097

## A 3D isothermal model for nematic liquid crystals with delay terms

 1 Departmento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain 2 Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy 3 Istituto di Matematica Applicata e Tecnologie Informatiche, CNR, Via Ferrata 1, 27100 Pavia, Italy

Dedicated to Maurizio Grasselli on the occasion of his 60th birthday

Received  December 2021 Revised  March 2022 Published  August 2022 Early access  April 2022

Fund Project: T. Caraballo was partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) and FEDER under projects US-1254251 and P18-FR-4509.
C. Cavaterra was partially supported by MIUR-PRIN Grant 2020F3NCPX "Mathematics for industry 4.0 (Math4I4)" and by GNAMPA "Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni" of INdAM "Istituto Nazionale di Alta Matematica".

In this paper we consider a model describing the evolution of a nematic liquid crystal flow with delay external forces. We analyze the evolution of the velocity field ${\boldsymbol u}$ which is ruled by the 3D incompressible Navier-Stokes system containing a delay term and with a stress tensor expressing the coupling between the transport and the induced terms. The dynamics of the director field $\boldsymbol{d}$ is described by a modified Allen-Cahn equation with a suitable penalization of the physical constraint $| \boldsymbol{d}| = 1$. We prove the existence of global in time weak solutions under appropriate assumptions, which in some cases requires the delay term to be small with respect to the viscosity parameter.

Citation: Tomás Caraballo, Cecilia Cavaterra. A 3D isothermal model for nematic liquid crystals with delay terms. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2117-2133. doi: 10.3934/dcdss.2022097
##### References:
 [1] T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807. [2] C. Cavaterra and E. Rocca, On a 3D isothermal model for nematic liquid crystals accounting for stretching terms, Z. Angew. Math. Phys., 64 (2013), 69-82.  doi: 10.1007/s00033-012-0219-7. [3] B. Climent-Ezquerra, F. Guillén-González and M. A. Rodríguez-Bellido, Stability for nematic liquid crystals with stretching terms, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2937-2942.  doi: 10.1142/S0218127410027477. [4] D. Coutand and S. Shkoller, Well posedness of the full Ericksen-Leslie model of nematic liquid crystals, C. R. Acad. Sci. Paris. Sér. I, 333 (2001), 919-924.  doi: 10.1016/S0764-4442(01)02161-9. [5] J. L. Ericksen, Equilibrium theory of liquid crystals, Advances in Liquid Crystals, 2 (1976), 233-398.  doi: 10.1016/B978-0-12-025002-8.50012-9. [6] E. Feireisl, M. Frémond, E. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Rational Mech. Anal., 205 (2012), 651-672.  doi: 10.1007/s00205-012-0517-4. [7] E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0. [8] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357.  doi: 10.1515/ans-2013-0205. [9] M. Grasselli and H. Wu, Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal flow, Z. Angew. Math. Phys., 62 (2011), 979-992.  doi: 10.1007/s00033-011-0157-9. [10] G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Roy. Soc. Proc., 102 (1922), 161-179. [11] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, Vol. 2 Gordon and Breach Science Publishers, New York-London-Paris, 1969. [12] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354. [13] F. M. Leslie, Theory of flow phenomena in liquid crystals, Advances in Liquid Crystals, 4 (1978), 1-81.  doi: 10.1016/B978-0-12-025004-2.50008-9. [14] 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. [15] F.-H. Lin and C. Liu, Static and dynamic theories of liquid crystals, J. Partial Differential Equations, 14 (2001), 289-330. [16] J.-L. Lions, Sur certaines équations paraboliques non linéaires, Bull. Soc. Math. France, 93 (1965), 155-175. [17] J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod, Paris, Gauthier-Villars, Paris, 1969. [18] C. Liu and J. Shen, On liquid crystal flows with free-slip boundary conditions, Discrete Contin. Dynam. Systems, 7 (2001), 307-318.  doi: 10.3934/dcds.2001.7.307. [19] L. Liu, T. Caraballo and P. Marín-Rubio, Stability results for 2D Navier–Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008. [20] J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, Teubner Texts in Mathematics, 52. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1983. [21] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. [22] H. Petzeltová, E. Rocca and G. Schimperna, On the long-time behavior of some mathematical models for nematic liquid crystals, Calc. Var. Partial Differential Equations, 46 (2013), 623-639.  doi: 10.1007/s00526-012-0496-1. [23] A. Segatti and H. Wu, Finite dimensional reduction and convergence to equilibrium for incompressible Smectic-A liquid crystal flows, Siam J. Math. Anal., 43 (2011), 2445-2481.  doi: 10.1137/100813427. [24] S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Comm. Part. Diff. Eq., 27 (2002), 1103-1137.  doi: 10.1081/PDE-120004895. [25] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [26] H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows, Discrete Contin. Dynam. Systems, 23 (2009), 455-475.  doi: 10.3934/dcds.2009.23.455. [27] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343. [28] H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345.  doi: 10.1007/s00526-011-0460-5. [29] J. H. Xu, T. Caraballo and J. Valero, Asymptotic behavior of nonlocal partial differential equations with long time memory, Discrete and Continuous Dynamical Systems, Series S, (2022). [30] J. H. Xu, Z. Zhang and T. Caraballo, Non-autonomous nonlocal partial differential equations with delay and memory, J. Differential Equations, 270 (2021), 505-546.  doi: 10.1016/j.jde.2020.07.037.

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##### References:
 [1] T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807. [2] C. Cavaterra and E. Rocca, On a 3D isothermal model for nematic liquid crystals accounting for stretching terms, Z. Angew. Math. Phys., 64 (2013), 69-82.  doi: 10.1007/s00033-012-0219-7. [3] B. Climent-Ezquerra, F. Guillén-González and M. A. Rodríguez-Bellido, Stability for nematic liquid crystals with stretching terms, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2937-2942.  doi: 10.1142/S0218127410027477. [4] D. Coutand and S. Shkoller, Well posedness of the full Ericksen-Leslie model of nematic liquid crystals, C. R. Acad. Sci. Paris. Sér. I, 333 (2001), 919-924.  doi: 10.1016/S0764-4442(01)02161-9. [5] J. L. Ericksen, Equilibrium theory of liquid crystals, Advances in Liquid Crystals, 2 (1976), 233-398.  doi: 10.1016/B978-0-12-025002-8.50012-9. [6] E. Feireisl, M. Frémond, E. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Rational Mech. Anal., 205 (2012), 651-672.  doi: 10.1007/s00205-012-0517-4. [7] E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0. [8] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357.  doi: 10.1515/ans-2013-0205. [9] M. Grasselli and H. Wu, Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal flow, Z. Angew. Math. Phys., 62 (2011), 979-992.  doi: 10.1007/s00033-011-0157-9. [10] G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Roy. Soc. Proc., 102 (1922), 161-179. [11] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, Vol. 2 Gordon and Breach Science Publishers, New York-London-Paris, 1969. [12] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354. [13] F. M. Leslie, Theory of flow phenomena in liquid crystals, Advances in Liquid Crystals, 4 (1978), 1-81.  doi: 10.1016/B978-0-12-025004-2.50008-9. [14] 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. [15] F.-H. Lin and C. Liu, Static and dynamic theories of liquid crystals, J. Partial Differential Equations, 14 (2001), 289-330. [16] J.-L. Lions, Sur certaines équations paraboliques non linéaires, Bull. Soc. Math. France, 93 (1965), 155-175. [17] J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod, Paris, Gauthier-Villars, Paris, 1969. [18] C. Liu and J. Shen, On liquid crystal flows with free-slip boundary conditions, Discrete Contin. Dynam. Systems, 7 (2001), 307-318.  doi: 10.3934/dcds.2001.7.307. [19] L. Liu, T. Caraballo and P. Marín-Rubio, Stability results for 2D Navier–Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008. [20] J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, Teubner Texts in Mathematics, 52. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1983. [21] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. [22] H. Petzeltová, E. Rocca and G. Schimperna, On the long-time behavior of some mathematical models for nematic liquid crystals, Calc. Var. Partial Differential Equations, 46 (2013), 623-639.  doi: 10.1007/s00526-012-0496-1. [23] A. Segatti and H. Wu, Finite dimensional reduction and convergence to equilibrium for incompressible Smectic-A liquid crystal flows, Siam J. Math. Anal., 43 (2011), 2445-2481.  doi: 10.1137/100813427. [24] S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Comm. Part. Diff. Eq., 27 (2002), 1103-1137.  doi: 10.1081/PDE-120004895. [25] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [26] H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows, Discrete Contin. Dynam. Systems, 23 (2009), 455-475.  doi: 10.3934/dcds.2009.23.455. [27] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343. [28] H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345.  doi: 10.1007/s00526-011-0460-5. [29] J. H. Xu, T. Caraballo and J. Valero, Asymptotic behavior of nonlocal partial differential equations with long time memory, Discrete and Continuous Dynamical Systems, Series S, (2022). [30] J. H. Xu, Z. Zhang and T. Caraballo, Non-autonomous nonlocal partial differential equations with delay and memory, J. Differential Equations, 270 (2021), 505-546.  doi: 10.1016/j.jde.2020.07.037.
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