-
Previous Article
Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials
- DCDS-S Home
- This Issue
-
Next Article
Orthogonality of fluxes in general nonlinear reaction networks
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.
|
[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. |
[1] |
Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078 |
[2] |
Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237 |
[3] |
Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052 |
[4] |
Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269 |
[5] |
Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks and Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303 |
[6] |
Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 171-183. doi: 10.3934/dcds.2010.27.171 |
[7] |
Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157 |
[8] |
Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215 |
[9] |
Vena Pearl Bongolan-walsh, David Cheban, Jinqiao Duan. Recurrent motions in the nonautonomous Navier-Stokes system. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 255-262. doi: 10.3934/dcdsb.2003.3.255 |
[10] |
D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure and Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499 |
[11] |
Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073 |
[12] |
C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 |
[13] |
Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 |
[14] |
Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161 |
[15] |
Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159 |
[16] |
Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153 |
[17] |
Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185 |
[18] |
John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371 |
[19] |
Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595 |
[20] |
Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]