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Weak sequential stability for a nonlinear model of nematic electrolytes
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 |
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.
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. Google Scholar |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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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