American Institute of Mathematical Sciences

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June  2019, 24(6): 2657-2681. doi: 10.3934/dcdsb.2018269

Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains

Chia-Yu Hsieh ,

National Center for Theoretical Sciences, National Taiwan University, Taipei City 10617, Taiwan

* Corresponding author: Chia-Yu Hsieh

Received  January 2018 Revised  June 2018 Published  October 2018

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. [18]) and find a new way to transform the perturbed problem. By choosing a suitable weighted norm, we then derive the associated energy law which can be used to prove that the $H^{-1}_x$ norm of the solution of the perturbed problem decays exponentially in time with the exponent independent of $\varepsilon$ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound.

Keywords: Poisson-Nernst-Planck system, charge conserving Poisson-Boltzmann equation, variable dielectric, boundary layer solution, stability, exponential decay estimate.
Mathematics Subject Classification: Primary: 35M33; Secondary: 35B25, 35B45.
Citation: Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Eisenberg, Ionic channels in biological membranes: Natural nanotubes, Acc. Chem. Res., 31 (1998), 117-123.  doi: 10.1021/ar950051e.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

B. Hille, Ion Channels of Excitable Membranes, 3rd edition, Sinauer Associates, Inc., 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. J. Hunter, Zeta Potential in Colloid Science, Academic Press Inc., 1981. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, SpringerVerlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions, Cambridge University Press, 1989. doi: 10.1017/CBO9780511608810.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Eisenberg, Ionic channels in biological membranes: Natural nanotubes, Acc. Chem. Res., 31 (1998), 117-123.  doi: 10.1021/ar950051e.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

B. Hille, Ion Channels of Excitable Membranes, 3rd edition, Sinauer Associates, Inc., 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. J. Hunter, Zeta Potential in Colloid Science, Academic Press Inc., 1981. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, SpringerVerlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions, Cambridge University Press, 1989. doi: 10.1017/CBO9780511608810.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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