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January  2021, 14(1): 219-241. doi: 10.3934/dcdss.2020366

Weak sequential stability for a nonlinear model of nematic electrolytes

Eduard Feireisl 1, , Elisabetta Rocca 2,, , Giulio Schimperna 2, and  Arghir Zarnescu 3,4,5,

1. 

Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Praha 1, Czech Republic
2. 

Università degli Studi di Pavia, Dipartimento di Matematica and IMATI-C.N.R, Via Ferrata 5, 27100, Pavia, Italy
3. 

IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Bizkaia, Spain
4. 

BCAM, Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Bizkaia, Spain
5. 

"Simion Stoilow" Institute of the Romanian Academy, 21 Calea Griviţei, 010702 Bucharest, Romania

* Corresponding author: Elisabetta Rocca

To Alex Mielke, with friendship and admiration

Received  September 2019 Revised  February 2020 Published  January 2021 Early access  May 2020

In this article we study a system of nonlinear PDEs modelling the electrokinetics of a nematic electrolyte material consisting of various ions species contained in a nematic liquid crystal.

The evolution is described by a system coupling a Nernst-Planck system for the ions concentrations with a Maxwell's equation of electrostatics governing the evolution of the electrostatic potential, a Navier-Stokes equation for the velocity field, and a non-smooth Allen-Cahn type equation for the nematic director field.

We focus on the two-species case and prove apriori estimates that provide a weak sequential stability result, the main step towards proving the existence of weak solutions.

Keywords: Nernst-Planck equation, Navier-Stokes system, a-priori estimates, compactness, weak-sequential stability.
Mathematics Subject Classification: Primary: 35Q60, 76D05; Secondary: 35B45.
Citation: Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
References:
[1]

R. Barberi, F. Ciuchi, G. E. Durand, M. Iovane, D. Sikharulidze, A. M. Sonnet and E. Virga, Electric field induced order reconstruction in a nematic cell, The European Physical Journal E, 13 (2004), 61-71.  doi: 10.1140/epje/e2004-00040-5.

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

[3]

D. Bothe, A. Fischer and J. Saal, Global well-posedness and stability of electrokinetic flows, SIAM J. Math. Anal., 46 (2014), 1263-1316.  doi: 10.1137/120880926.

[4]

M. C. Calderer, D. Golovaty, O. Lavrentovich and J. N. Walkington, Modeling of nematic electrolytes and nonlinear electroosmosis, SIAM J. Appl. Math., 76 (2016), 2260-2285.  doi: 10.1137/16M1056225.

[5]

C. Cavaterra, E. Rocca and H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows, J. Differential Equations, 255 (2013), 24-57.  doi: 10.1016/j.jde.2013.03.009.

[6]

G. Cimatti and I. Fragalà, Invariant regions for the Nernst-Planck equations, Ann. Mat. Pura Appl., 175 (1998), 93-118.  doi: 10.1007/BF01783677.

[7]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112.  doi: 10.3934/mcrf.2016.6.95.

[8]

P. Constantin and M. Ignatova, On the Nernst-Planck-Navier-Stokes system, Arch Rational Mech Anal, 232 (2019), 1379-1428.  doi: 10.1007/s00205-018-01345-6.

[9]

J. L. Ericksen, Conservation laws For liquid crystals, Trans. Soc. Rheol., 5 (1961), 23-34.  doi: 10.1122/1.548883.

[10]

J. L. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392. 

[11]

J. L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal., 113 (1990), 97-120. 

[12]

E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Commun. Math. Sci., 12 (2014), 317-343.  doi: 10.4310/CMS.2014.v12.n2.a6.

[13]

E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy, Ann. Mat. Pura Appl., 194 (2015), 1269-1299.  doi: 10.1007/s10231-014-0419-1.

[14]

A. Fischer and J. Saal, Global weak solutions in three space dimensions for electrokinetic flow processes, J. Evol. Equ., 17 (2017), 309-333.  doi: 10.1007/s00028-016-0356-0.

[15]

H. Gong, C. Wang and X. Zhang, Partial regularity of suitable weak solutions of the Navier-Stokes-Planck-Nernst-Poisson equation, arXiv: 1905.13365.

[16]

F. M. Leslie, Theory of flow phenomenum in liquid crystals, Brown (Ed.), A.P., New York, 4 (1979), 1–81.

[17]

F. M. Leslie, Continuum theory for nematic liquid crystals, Contin. Mech. Thermodyn, 4 (1992), 167-175.  doi: 10.1007/BF01130288.

[18]

F. Lin, On nematic liquid crystals with variable degree of orientation, Comm. Pure Appl. Math., 44 (1991), 453-468.  doi: 10.1002/cpa.3160440404.

[19]

F. Lin and C. Liu, Static and dynamic theories of liquid crystals, J. Partial Differential Equations, 14 (2001), 289-330. 

[20]

F. Lin and C. Wang, The Analysis of Harmonic Maps and Their Heat Flows, World Scientific, 2008. doi: 10.1142/9789812779533.

[21]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp. doi: 10.1098/rsta.2013.0361.

[22]

N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206. 

[23]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. 

[24]

R. Nochetto, S. Walker and W. Zhang, A finite element method for nematic liquid crystals with variable degree of orientation, SIAM J. Numer. Anal., 55 (2017), 1357-1386.  doi: 10.1137/15M103844X.

[25]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1015.  doi: 10.1142/S0218202509003693.

[26]

O. M. Tovkach, C. Conklin, M. C. Calderer, D. Golovaty, O. Lavrentovich, J. Viñals and N. J. Walkington, Q-tensor model for electrokinetics in nematic liquid crystals, Phys. Rev. Fluids, 2 (2017), 053302. doi: 10.1103/PhysRevFluids.2.053302.

[27]

E. G. Virga, Variational Theories for Liquid Crystals, Applied Mathematics and Mathematical Computation 8, Chapman & Hall, London, 1994.

show all references

References:
[1]

R. Barberi, F. Ciuchi, G. E. Durand, M. Iovane, D. Sikharulidze, A. M. Sonnet and E. Virga, Electric field induced order reconstruction in a nematic cell, The European Physical Journal E, 13 (2004), 61-71.  doi: 10.1140/epje/e2004-00040-5.

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

[3]

D. Bothe, A. Fischer and J. Saal, Global well-posedness and stability of electrokinetic flows, SIAM J. Math. Anal., 46 (2014), 1263-1316.  doi: 10.1137/120880926.

[4]

M. C. Calderer, D. Golovaty, O. Lavrentovich and J. N. Walkington, Modeling of nematic electrolytes and nonlinear electroosmosis, SIAM J. Appl. Math., 76 (2016), 2260-2285.  doi: 10.1137/16M1056225.

[5]

C. Cavaterra, E. Rocca and H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows, J. Differential Equations, 255 (2013), 24-57.  doi: 10.1016/j.jde.2013.03.009.

[6]

G. Cimatti and I. Fragalà, Invariant regions for the Nernst-Planck equations, Ann. Mat. Pura Appl., 175 (1998), 93-118.  doi: 10.1007/BF01783677.

[7]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112.  doi: 10.3934/mcrf.2016.6.95.

[8]

P. Constantin and M. Ignatova, On the Nernst-Planck-Navier-Stokes system, Arch Rational Mech Anal, 232 (2019), 1379-1428.  doi: 10.1007/s00205-018-01345-6.

[9]

J. L. Ericksen, Conservation laws For liquid crystals, Trans. Soc. Rheol., 5 (1961), 23-34.  doi: 10.1122/1.548883.

[10]

J. L. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392. 

[11]

J. L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal., 113 (1990), 97-120. 

[12]

E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Commun. Math. Sci., 12 (2014), 317-343.  doi: 10.4310/CMS.2014.v12.n2.a6.

[13]

E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy, Ann. Mat. Pura Appl., 194 (2015), 1269-1299.  doi: 10.1007/s10231-014-0419-1.

[14]

A. Fischer and J. Saal, Global weak solutions in three space dimensions for electrokinetic flow processes, J. Evol. Equ., 17 (2017), 309-333.  doi: 10.1007/s00028-016-0356-0.

[15]

H. Gong, C. Wang and X. Zhang, Partial regularity of suitable weak solutions of the Navier-Stokes-Planck-Nernst-Poisson equation, arXiv: 1905.13365.

[16]

F. M. Leslie, Theory of flow phenomenum in liquid crystals, Brown (Ed.), A.P., New York, 4 (1979), 1–81.

[17]

F. M. Leslie, Continuum theory for nematic liquid crystals, Contin. Mech. Thermodyn, 4 (1992), 167-175.  doi: 10.1007/BF01130288.

[18]

F. Lin, On nematic liquid crystals with variable degree of orientation, Comm. Pure Appl. Math., 44 (1991), 453-468.  doi: 10.1002/cpa.3160440404.

[19]

F. Lin and C. Liu, Static and dynamic theories of liquid crystals, J. Partial Differential Equations, 14 (2001), 289-330. 

[20]

F. Lin and C. Wang, The Analysis of Harmonic Maps and Their Heat Flows, World Scientific, 2008. doi: 10.1142/9789812779533.

[21]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp. doi: 10.1098/rsta.2013.0361.

[22]

N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206. 

[23]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. 

[24]

R. Nochetto, S. Walker and W. Zhang, A finite element method for nematic liquid crystals with variable degree of orientation, SIAM J. Numer. Anal., 55 (2017), 1357-1386.  doi: 10.1137/15M103844X.

[25]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1015.  doi: 10.1142/S0218202509003693.

[26]

O. M. Tovkach, C. Conklin, M. C. Calderer, D. Golovaty, O. Lavrentovich, J. Viñals and N. J. Walkington, Q-tensor model for electrokinetics in nematic liquid crystals, Phys. Rev. Fluids, 2 (2017), 053302. doi: 10.1103/PhysRevFluids.2.053302.

[27]

E. G. Virga, Variational Theories for Liquid Crystals, Applied Mathematics and Mathematical Computation 8, Chapman & Hall, London, 1994.

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