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Homotopy invariants methods in the global dynamics of strongly damped wave equation
Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients
1. | Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu City 30014, Taiwan |
References:
[1] |
V. Barcilon, D.-P. Chen, R. S. Eisenberg and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study, SIAM J. Appl. Math., 57 (1997), 631-648.
doi: 10.1137/S0036139995312149. |
[2] |
M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 70 (2004), 021506.
doi: 10.1103/PhysRevE.70.021506. |
[3] |
M. Z. Bazant, K. T. Chu and B. J. Bayly, Current-Voltage relations for electrochemical thin films, SIAM J. Appl. Math., 65 (2005), 1463-1484.
doi: 10.1137/040609938. |
[4] |
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. |
[5] |
S. L. Carnie, D. Y. C. Chan and J. Stankovich, Computation of forces between spherical colloidal particles: Nonlinear Poisson-Boltzmann theory, J. Colloid Interface Sci., 165 (1994), 116-128.
doi: 10.1006/jcis.1994.1212. |
[6] |
Y. S. Choi and R. Lui, An integro-differential equation arising from an electrochemistry model, Quart. Appl. Math., 55 (1997), 677-686. |
[7] |
D. T. Conroy, R. V. Craster, O. K. Matar and D. T. Papageorgiou, Dynamics and stability of an annular electrolyte film, J. Fluid Mech., 656 (2010), 481-506.
doi: 10.1017/S0022112010001254. |
[8] |
B. Eisenberg, Ionic channels in biological membranes: Natural nanotubes, Acc. Chem. Res., 31 (1998), 117-123.
doi: 10.1021/ar950051e. |
[9] |
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions II, Based, in part, on notes left by Harry Bateman, McGraw-Hill, New York, 1953. Available from: http://authors.library.caltech.edu/43491/. |
[10] |
W. Fang and K. Ito, Existence and Uniqueness of Steady-State Solutions for an Electrochemistry Model, P. Am. Math. Soc., 129 (2001), 1037-1040.
doi: 10.1090/S0002-9939-00-05769-5. |
[11] |
M. A. Fontelos and L. B. Gamboa, On the structure of double layers in Poisson-Boltzmann equation, Discrete Contin Dyn. Syst. B, 17 (2012), 1939-1967.
doi: 10.3934/dcdsb.2012.17.1939. |
[12] |
A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the Poisson-Boltzmann equation, J. Differential Equations, 69 (1987), 15-38.
doi: 10.1016/0022-0396(87)90100-8. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, 2001.
doi: 10.1007/978-3-642-61798-0. |
[14] |
A. Glitzky and R. Hünlich, Energetic estimates and asymptotics for electro-reaction-diffusion systems, Z. Angew. Math. Mech., 77 (1997), 823-832.
doi: 10.1002/zamm.19970771105. |
[15] |
M. J. Holst, Multilevel Methods for the Poisson-Boltzmann Equation, Ph.D thesis, Numerical Computing Group, University of Illinois at Urbana-Champaign, 1993. Available from: http://ccom.ucsd.edu/~mholst/pubs/dist/Hols94d.pdf. |
[16] |
Y. Hyon, A Mathematical Model For Electrical Activity in Cell Membrane: Energetic Variational Approach,, work in progress., ().
|
[17] |
Y. J. Kang, C. Yang and X. Y. Huang, Electroosmotic flow in a capillary annulus with high zeta potentials, J. Colloid Interface Sci., 253 (2002), 285-294.
doi: 10.1006/jcis.2002.8453. |
[18] |
C. Koch, Biophysics of Computation, Oxford University Press, Canada, 1999. Available from: http://www.oupcanada.com/catalog/9780195181999.html. |
[19] |
D. Lacoste, G. I. Menon, M. Z. Bazant and J. F. Joanny, Electrostatic and electrokinetic contributions to the elastic moduli of a driven membrane, Eur. Phys. J. E, 28 (2009), 243-264.
doi: 10.1140/epje/i2008-10433-1. |
[20] |
L. Lanzani and Z. Shen, On the Robin boundary condition for Laplace's equation in Lipschit domains, Commun. Partial Differ. Eq., 29 (2004), 91-109.
doi: 10.1081/PDE-120028845. |
[21] |
C. C. Lee, The charge conserving Poisson-Boltzmann equations: Existence, uniqueness and maximum principle, J. Math. Phys., 55 (2014), 051503, 16pp.
doi: 10.1063/1.4878492. |
[22] |
C. C. Lee, H. Lee, Y. Hyon, T. C. Lin and C. Liu, New Poisson-Boltzmann type equations: One-dimensional solutions, Nonlinearity, 24 (2011), 431-458.
doi: 10.1088/0951-7715/24/2/004. |
[23] |
C. C. Lee, H. Lee, Y. Hyon, T. C. Lin and C. Liu, Boundary layer solutions of Charge Conserving Poisson-Boltzmann equations: One-dimensional case,, to appear in Commun. Math. Sci., ().
|
[24] |
W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differential Equations, 246 (2009), 428-451.
doi: 10.1016/j.jde.2008.09.010. |
[25] |
W. Liu and H. Xu, A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow, J. Differential Equations, 258 (2015), 1192-1228.
doi: 10.1016/j.jde.2014.10.015. |
[26] |
W. Nonner and B. Eisenberg, Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels, Biophys. J., 75 (1998), 1287-1305.
doi: 10.1016/S0006-3495(98)74048-2. |
[27] |
J. H. Park and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609-630.
doi: 10.1137/S0036139995279809. |
[28] |
I. Rubinstein, Counterion condensation as an exact limiting property of solutions of the Poisson-Boltzmann equation, SIAM J. Appl. Math., 46 (1986), 1024-1038.
doi: 10.1137/0146061. |
[29] |
R. Ryham, C. Liu and Z. Q. Wang, On electro-kinetic fluids: One dimensional configurations, Discrete Contin Dyn. Syst. B, 6 (2006), 357-371.
doi: 10.3934/dcdsb.2006.6.357. |
[30] |
R. Ryham, C. Liu and L. Zikatanov, Mathematical models for the deformation of electrolyte droplets, Discrete Contin Dyn. Syst. B, 8 (2007), 649-661.
doi: 10.3934/dcdsb.2007.8.649. |
[31] |
H. Sugioka, Ion-conserving Poisson-Boltzmann theory, Phys. Rev. E, 86 (2012), 016318.
doi: 10.1103/PhysRevE.86.016318. |
[32] |
L. Wan, S. Xu, M. Liao, C. Liu and P. Sheng, Self-consistent approach to global charge neutrality in electrokinetics: A surface potential trap model, Phys. Rev. X, 4 (2014), 011042.
doi: 10.1103/PhysRevX.4.011042. |
[33] |
F. Ziebert, M. Z. Bazant and D. Lacoste, Effective zero-thickness model for a conductive membrane driven by an electric field, Phys. Rev. E, 81 (2010), 031912.
doi: 10.1103/PhysRevE.81.031912. |
[34] |
S. Zhou, Z. Wang and B. Li, Mean-field description of ionic size effects with non-uniform ionic sizes: A numerical approach, Phys. Rev. E, 84 (2011), 021901.
doi: 10.1103/PhysRevE.84.021901. |
show all references
References:
[1] |
V. Barcilon, D.-P. Chen, R. S. Eisenberg and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study, SIAM J. Appl. Math., 57 (1997), 631-648.
doi: 10.1137/S0036139995312149. |
[2] |
M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 70 (2004), 021506.
doi: 10.1103/PhysRevE.70.021506. |
[3] |
M. Z. Bazant, K. T. Chu and B. J. Bayly, Current-Voltage relations for electrochemical thin films, SIAM J. Appl. Math., 65 (2005), 1463-1484.
doi: 10.1137/040609938. |
[4] |
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. |
[5] |
S. L. Carnie, D. Y. C. Chan and J. Stankovich, Computation of forces between spherical colloidal particles: Nonlinear Poisson-Boltzmann theory, J. Colloid Interface Sci., 165 (1994), 116-128.
doi: 10.1006/jcis.1994.1212. |
[6] |
Y. S. Choi and R. Lui, An integro-differential equation arising from an electrochemistry model, Quart. Appl. Math., 55 (1997), 677-686. |
[7] |
D. T. Conroy, R. V. Craster, O. K. Matar and D. T. Papageorgiou, Dynamics and stability of an annular electrolyte film, J. Fluid Mech., 656 (2010), 481-506.
doi: 10.1017/S0022112010001254. |
[8] |
B. Eisenberg, Ionic channels in biological membranes: Natural nanotubes, Acc. Chem. Res., 31 (1998), 117-123.
doi: 10.1021/ar950051e. |
[9] |
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions II, Based, in part, on notes left by Harry Bateman, McGraw-Hill, New York, 1953. Available from: http://authors.library.caltech.edu/43491/. |
[10] |
W. Fang and K. Ito, Existence and Uniqueness of Steady-State Solutions for an Electrochemistry Model, P. Am. Math. Soc., 129 (2001), 1037-1040.
doi: 10.1090/S0002-9939-00-05769-5. |
[11] |
M. A. Fontelos and L. B. Gamboa, On the structure of double layers in Poisson-Boltzmann equation, Discrete Contin Dyn. Syst. B, 17 (2012), 1939-1967.
doi: 10.3934/dcdsb.2012.17.1939. |
[12] |
A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the Poisson-Boltzmann equation, J. Differential Equations, 69 (1987), 15-38.
doi: 10.1016/0022-0396(87)90100-8. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, 2001.
doi: 10.1007/978-3-642-61798-0. |
[14] |
A. Glitzky and R. Hünlich, Energetic estimates and asymptotics for electro-reaction-diffusion systems, Z. Angew. Math. Mech., 77 (1997), 823-832.
doi: 10.1002/zamm.19970771105. |
[15] |
M. J. Holst, Multilevel Methods for the Poisson-Boltzmann Equation, Ph.D thesis, Numerical Computing Group, University of Illinois at Urbana-Champaign, 1993. Available from: http://ccom.ucsd.edu/~mholst/pubs/dist/Hols94d.pdf. |
[16] |
Y. Hyon, A Mathematical Model For Electrical Activity in Cell Membrane: Energetic Variational Approach,, work in progress., ().
|
[17] |
Y. J. Kang, C. Yang and X. Y. Huang, Electroosmotic flow in a capillary annulus with high zeta potentials, J. Colloid Interface Sci., 253 (2002), 285-294.
doi: 10.1006/jcis.2002.8453. |
[18] |
C. Koch, Biophysics of Computation, Oxford University Press, Canada, 1999. Available from: http://www.oupcanada.com/catalog/9780195181999.html. |
[19] |
D. Lacoste, G. I. Menon, M. Z. Bazant and J. F. Joanny, Electrostatic and electrokinetic contributions to the elastic moduli of a driven membrane, Eur. Phys. J. E, 28 (2009), 243-264.
doi: 10.1140/epje/i2008-10433-1. |
[20] |
L. Lanzani and Z. Shen, On the Robin boundary condition for Laplace's equation in Lipschit domains, Commun. Partial Differ. Eq., 29 (2004), 91-109.
doi: 10.1081/PDE-120028845. |
[21] |
C. C. Lee, The charge conserving Poisson-Boltzmann equations: Existence, uniqueness and maximum principle, J. Math. Phys., 55 (2014), 051503, 16pp.
doi: 10.1063/1.4878492. |
[22] |
C. C. Lee, H. Lee, Y. Hyon, T. C. Lin and C. Liu, New Poisson-Boltzmann type equations: One-dimensional solutions, Nonlinearity, 24 (2011), 431-458.
doi: 10.1088/0951-7715/24/2/004. |
[23] |
C. C. Lee, H. Lee, Y. Hyon, T. C. Lin and C. Liu, Boundary layer solutions of Charge Conserving Poisson-Boltzmann equations: One-dimensional case,, to appear in Commun. Math. Sci., ().
|
[24] |
W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differential Equations, 246 (2009), 428-451.
doi: 10.1016/j.jde.2008.09.010. |
[25] |
W. Liu and H. Xu, A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow, J. Differential Equations, 258 (2015), 1192-1228.
doi: 10.1016/j.jde.2014.10.015. |
[26] |
W. Nonner and B. Eisenberg, Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels, Biophys. J., 75 (1998), 1287-1305.
doi: 10.1016/S0006-3495(98)74048-2. |
[27] |
J. H. Park and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609-630.
doi: 10.1137/S0036139995279809. |
[28] |
I. Rubinstein, Counterion condensation as an exact limiting property of solutions of the Poisson-Boltzmann equation, SIAM J. Appl. Math., 46 (1986), 1024-1038.
doi: 10.1137/0146061. |
[29] |
R. Ryham, C. Liu and Z. Q. Wang, On electro-kinetic fluids: One dimensional configurations, Discrete Contin Dyn. Syst. B, 6 (2006), 357-371.
doi: 10.3934/dcdsb.2006.6.357. |
[30] |
R. Ryham, C. Liu and L. Zikatanov, Mathematical models for the deformation of electrolyte droplets, Discrete Contin Dyn. Syst. B, 8 (2007), 649-661.
doi: 10.3934/dcdsb.2007.8.649. |
[31] |
H. Sugioka, Ion-conserving Poisson-Boltzmann theory, Phys. Rev. E, 86 (2012), 016318.
doi: 10.1103/PhysRevE.86.016318. |
[32] |
L. Wan, S. Xu, M. Liao, C. Liu and P. Sheng, Self-consistent approach to global charge neutrality in electrokinetics: A surface potential trap model, Phys. Rev. X, 4 (2014), 011042.
doi: 10.1103/PhysRevX.4.011042. |
[33] |
F. Ziebert, M. Z. Bazant and D. Lacoste, Effective zero-thickness model for a conductive membrane driven by an electric field, Phys. Rev. E, 81 (2010), 031912.
doi: 10.1103/PhysRevE.81.031912. |
[34] |
S. Zhou, Z. Wang and B. Li, Mean-field description of ionic size effects with non-uniform ionic sizes: A numerical approach, Phys. Rev. E, 84 (2011), 021901.
doi: 10.1103/PhysRevE.84.021901. |
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