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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  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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.   Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.   Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

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