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Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study
Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains
National Center for Theoretical Sciences, National Taiwan University, Taipei City 10617, Taiwan |
In this paper, we consider radial solutions of the Poisson-Nernst-Planck (PNP) system with variable dielectric coefficients $\varepsilon g(x)$ in $N$-dimensional annular domains, $N≥2$. When the parameter $\varepsilon$ tends to zero, the PNP system admits a boundary layer solution as a steady state, which satisfies the charge conserving Poisson-Boltzmann (CCPB) equation. For the stability of the radial boundary layer solutions to the time-dependent radial PNP system, we estimate the radial solution of the perturbed problem with global electroneutrality. We generalize the argument of the one spatial dimension case (cf. [
References:
[1] |
G. Allaire, J.-F. Dufrêche, A. Mikelić and A. Piatnitski,
Asymptotic analysis of the Poisson-Boltzmann equation describing electrokinetics in porous media, Nonlinearity, 26 (2013), 881-910.
doi: 10.1088/0951-7715/26/3/881. |
[2] |
A. Arnold, P. Markowich and G. Toscani,
On large time asymptotics for drift-diffusion Poisson systems, Transport Theory Statist. Phys., 29 (2000), 571-581.
doi: 10.1080/00411450008205893. |
[3] |
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. |
[4] |
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. |
[5] |
P. Biler and J. Dolbeault,
Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré, 1 (2000), 461-472.
doi: 10.1007/s000230050003. |
[6] |
P. Biler, W. Hebisch and T. Nadzieja,
The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[7] |
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. |
[8] |
J. Cartailler, Z. Schuss and D. Holcman,
Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the Voltage-Current relation in neurobiological microdomains, Phys. D, 339 (2017), 39-48.
doi: 10.1016/j.physd.2016.09.001. |
[9] |
D. Chen, J. Lear and B. Eisenberg,
Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel, Biophys J., 72 (1997), 97-116.
doi: 10.1016/S0006-3495(97)78650-8. |
[10] |
B. Eisenberg,
Ionic channels in biological membranes: Natural nanotubes, Acc. Chem. Res., 31 (1998), 117-123.
doi: 10.1021/ar950051e. |
[11] |
B. Eisenberg and W. Liu,
Poisson-Nernst-Planck systems for ion channels with permanent charges, SIAM J. Math. Anal., 38 (2007), 1932-1966.
doi: 10.1137/060657480. |
[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] |
H. Gajewski,
On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors, Z. Angew. Math. Mech., 65 (1985), 101-108.
doi: 10.1002/zamm.19850650210. |
[14] |
H. Gajewski and K. Gröger,
On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl., 113 (1986), 12-35.
doi: 10.1016/0022-247X(86)90330-6. |
[15] |
D. Gillespie, W. Nonner and R. S. Eisenberg,
Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux, J. Phys.: Condens. Matter, 14 (2002), 12129-12145.
doi: 10.1088/0953-8984/14/46/317. |
[16] |
B. Hille, Ion Channels of Excitable Membranes, 3rd edition, Sinauer Associates, Inc., 2001. |
[17] |
C.-Y. Hsieh, Y. Hyon, H. Lee, T.-C. Lin and C. Liu,
Transport of charged particles: Entropy production and maximum dissipation principle, J. Math. Anal. Appl., 422 (2015), 309-336.
doi: 10.1016/j.jmaa.2014.07.078. |
[18] |
C.-Y. Hsieh and T.-C. Lin,
Exponential decay estimates for the stability of boundary layer solutions to Poisson-Nernst-Planck systems: one spatial dimension case, SIAM J. Math. Anal., 47 (2015), 3442-3465.
doi: 10.1137/140994095. |
[19] |
R. J. Hunter, Zeta Potential in Colloid Science, Academic Press Inc., 1981. |
[20] |
M. S. Kilic, M. Z. Bazant and A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. Ⅰ. Double-layer charging, Phys. Rev. E, 75 (2007), 021502.
doi: 10.1103/PhysRevE.75.021502. |
[21] |
M. S. Kilic, M. Z. Bazant and A. Ajdari,
Steric effects in the dynamics of electrolytes at large applied voltages. Ⅱ. Modified Poisson-Nernst-Planck equations, Phys. Rev. E, 75 (2007), 021503.
doi: 10.1103/PhysRevE.75.021503. |
[22] |
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. |
[23] |
C.-C. Lee,
Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients, Discrete Contin. Dyn. Syst., 36 (2016), 3251-3276.
doi: 10.3934/dcds.2016.36.3251. |
[24] |
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. |
[25] |
C.-C. Lee, T.-C. Lin and J.-H. Lyu, Boundary layer solutions of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients for radially symmetric case, preprint. |
[26] |
T.-C. Lin and B. Eisenberg,
A new approach to the Lennard-Jones potential and a new model: PNP-steric equations, Commun. Math. Sci., 12 (2014), 149-173.
doi: 10.4310/CMS.2014.v12.n1.a7. |
[27] |
P. Liu, X. Ji and Z. Xu,
Modified Poisson-Nernst-Planck model with accurate Coulomb correlation in variable media, SIAM J. Appl. Math., 78 (2018), 226-245.
doi: 10.1137/16M110383X. |
[28] |
W. Liu,
Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems, SIAM J. Appl. Math., 65 (2005), 754-766.
doi: 10.1137/S0036139903420931. |
[29] |
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, SpringerVerlag, Vienna, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[30] |
Y. Mori, J. W. Jerome and C. S. Peskin,
A three-dimensional model of cellular electrical activity, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 367-390.
|
[31] |
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. |
[32] |
O. J. Riveros, T. L. Croxton and W. M. Armstrong,
Liquid junction potentials calculated from numerical solutions of the Nernst-Planck and Poisson equations, J. Theor. Biol., 140 (1989), 221-230.
doi: 10.1016/S0022-5193(89)80130-4. |
[33] |
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. |
[34] |
W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions, Cambridge University Press, 1989.
doi: 10.1017/CBO9780511608810. |
[35] |
R. Ryham, C. Liu and Z.-Q. Wang,
On electro-kinetic fluids: One dimensional configurations, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 357-371.
|
[36] |
R. Ryham, C. Liu and L. Zikatanov,
Mathematical models for the deformation of electrolyte droplets, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 649-661.
doi: 10.3934/dcdsb.2007.8.649. |
[37] |
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. |
[38] |
Y. Wang, C. Liu and Z. Tan,
A generalized Poisson-Nernst-Planck-Navier-Stokes model on the fluid with the crowded charged particles: derivation and its well-posedness, SIAM J. Math. Anal., 48 (2016), 3191-3235.
doi: 10.1137/16M1055104. |
[39] |
S. Xu, P. Sheng and C. Liu,
An energetic variational approach for ion transport, Commun. Math. Sci., 12 (2014), 779-789.
doi: 10.4310/CMS.2014.v12.n4.a9. |
[40] |
J. Zhang, X. Gong, C. Liu, W. Wen and P. Sheng, Electrorheological fluid dynamics, Phys. Rev. Lett., 101 (2008), 194503.
doi: 10.1063/1.2897856. |
[41] |
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. |
show all references
References:
[1] |
G. Allaire, J.-F. Dufrêche, A. Mikelić and A. Piatnitski,
Asymptotic analysis of the Poisson-Boltzmann equation describing electrokinetics in porous media, Nonlinearity, 26 (2013), 881-910.
doi: 10.1088/0951-7715/26/3/881. |
[2] |
A. Arnold, P. Markowich and G. Toscani,
On large time asymptotics for drift-diffusion Poisson systems, Transport Theory Statist. Phys., 29 (2000), 571-581.
doi: 10.1080/00411450008205893. |
[3] |
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. |
[4] |
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. |
[5] |
P. Biler and J. Dolbeault,
Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré, 1 (2000), 461-472.
doi: 10.1007/s000230050003. |
[6] |
P. Biler, W. Hebisch and T. Nadzieja,
The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[7] |
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. |
[8] |
J. Cartailler, Z. Schuss and D. Holcman,
Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the Voltage-Current relation in neurobiological microdomains, Phys. D, 339 (2017), 39-48.
doi: 10.1016/j.physd.2016.09.001. |
[9] |
D. Chen, J. Lear and B. Eisenberg,
Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel, Biophys J., 72 (1997), 97-116.
doi: 10.1016/S0006-3495(97)78650-8. |
[10] |
B. Eisenberg,
Ionic channels in biological membranes: Natural nanotubes, Acc. Chem. Res., 31 (1998), 117-123.
doi: 10.1021/ar950051e. |
[11] |
B. Eisenberg and W. Liu,
Poisson-Nernst-Planck systems for ion channels with permanent charges, SIAM J. Math. Anal., 38 (2007), 1932-1966.
doi: 10.1137/060657480. |
[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] |
H. Gajewski,
On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors, Z. Angew. Math. Mech., 65 (1985), 101-108.
doi: 10.1002/zamm.19850650210. |
[14] |
H. Gajewski and K. Gröger,
On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl., 113 (1986), 12-35.
doi: 10.1016/0022-247X(86)90330-6. |
[15] |
D. Gillespie, W. Nonner and R. S. Eisenberg,
Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux, J. Phys.: Condens. Matter, 14 (2002), 12129-12145.
doi: 10.1088/0953-8984/14/46/317. |
[16] |
B. Hille, Ion Channels of Excitable Membranes, 3rd edition, Sinauer Associates, Inc., 2001. |
[17] |
C.-Y. Hsieh, Y. Hyon, H. Lee, T.-C. Lin and C. Liu,
Transport of charged particles: Entropy production and maximum dissipation principle, J. Math. Anal. Appl., 422 (2015), 309-336.
doi: 10.1016/j.jmaa.2014.07.078. |
[18] |
C.-Y. Hsieh and T.-C. Lin,
Exponential decay estimates for the stability of boundary layer solutions to Poisson-Nernst-Planck systems: one spatial dimension case, SIAM J. Math. Anal., 47 (2015), 3442-3465.
doi: 10.1137/140994095. |
[19] |
R. J. Hunter, Zeta Potential in Colloid Science, Academic Press Inc., 1981. |
[20] |
M. S. Kilic, M. Z. Bazant and A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. Ⅰ. Double-layer charging, Phys. Rev. E, 75 (2007), 021502.
doi: 10.1103/PhysRevE.75.021502. |
[21] |
M. S. Kilic, M. Z. Bazant and A. Ajdari,
Steric effects in the dynamics of electrolytes at large applied voltages. Ⅱ. Modified Poisson-Nernst-Planck equations, Phys. Rev. E, 75 (2007), 021503.
doi: 10.1103/PhysRevE.75.021503. |
[22] |
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. |
[23] |
C.-C. Lee,
Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients, Discrete Contin. Dyn. Syst., 36 (2016), 3251-3276.
doi: 10.3934/dcds.2016.36.3251. |
[24] |
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. |
[25] |
C.-C. Lee, T.-C. Lin and J.-H. Lyu, Boundary layer solutions of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients for radially symmetric case, preprint. |
[26] |
T.-C. Lin and B. Eisenberg,
A new approach to the Lennard-Jones potential and a new model: PNP-steric equations, Commun. Math. Sci., 12 (2014), 149-173.
doi: 10.4310/CMS.2014.v12.n1.a7. |
[27] |
P. Liu, X. Ji and Z. Xu,
Modified Poisson-Nernst-Planck model with accurate Coulomb correlation in variable media, SIAM J. Appl. Math., 78 (2018), 226-245.
doi: 10.1137/16M110383X. |
[28] |
W. Liu,
Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems, SIAM J. Appl. Math., 65 (2005), 754-766.
doi: 10.1137/S0036139903420931. |
[29] |
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, SpringerVerlag, Vienna, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[30] |
Y. Mori, J. W. Jerome and C. S. Peskin,
A three-dimensional model of cellular electrical activity, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 367-390.
|
[31] |
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. |
[32] |
O. J. Riveros, T. L. Croxton and W. M. Armstrong,
Liquid junction potentials calculated from numerical solutions of the Nernst-Planck and Poisson equations, J. Theor. Biol., 140 (1989), 221-230.
doi: 10.1016/S0022-5193(89)80130-4. |
[33] |
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. |
[34] |
W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions, Cambridge University Press, 1989.
doi: 10.1017/CBO9780511608810. |
[35] |
R. Ryham, C. Liu and Z.-Q. Wang,
On electro-kinetic fluids: One dimensional configurations, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 357-371.
|
[36] |
R. Ryham, C. Liu and L. Zikatanov,
Mathematical models for the deformation of electrolyte droplets, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 649-661.
doi: 10.3934/dcdsb.2007.8.649. |
[37] |
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. |
[38] |
Y. Wang, C. Liu and Z. Tan,
A generalized Poisson-Nernst-Planck-Navier-Stokes model on the fluid with the crowded charged particles: derivation and its well-posedness, SIAM J. Math. Anal., 48 (2016), 3191-3235.
doi: 10.1137/16M1055104. |
[39] |
S. Xu, P. Sheng and C. Liu,
An energetic variational approach for ion transport, Commun. Math. Sci., 12 (2014), 779-789.
doi: 10.4310/CMS.2014.v12.n4.a9. |
[40] |
J. Zhang, X. Gong, C. Liu, W. Wen and P. Sheng, Electrorheological fluid dynamics, Phys. Rev. Lett., 101 (2008), 194503.
doi: 10.1063/1.2897856. |
[41] |
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. |
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