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The Time Decay Rates of the Classical Solution to the Poisson-Nernst-Planck-Fourier Equations in ℝ3

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Abstract

In this work, the Poisson-Nernst-Planck-Fourier system in three dimensions is considered. For when the initial data regards a small perturbation around the constant equilibrium state in a H3 ∩ Ḣ−s (0 ≤ s ≤ 1/2) norm, we obtain the time convergence rate of the global solution by a regularity interpolation trick and an energy method.

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References

  1. Barcilon V, Chen D P, Eisenberg R S, Jerome J. Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study. SIAM J Appl Math, 1997, 57: 631–648

    Article  MathSciNet  MATH  Google Scholar 

  2. Biler P, Dolbeault J. Long Time Behavior of Solutions to Nernst-Planck and Debye-Hückel Drift-Diffusion Systems. Annales Henri Poincaré, 2000, 1: 461–472

    Article  MathSciNet  MATH  Google Scholar 

  3. Biler P, Hebisch W, Nadzieja T. The Debye system: existence and large time behavior of solutions. Nonlinear Anal, 1994, 23: 1189–1209

    Article  MathSciNet  MATH  Google Scholar 

  4. Bazant M Z, Thornton K, Ajdari A. Diffuse-charge dynamics in electrochemical systems. Physical review E, 2004, 70: 021,506

    Article  Google Scholar 

  5. Cesare P, Moriondo A, Vellani V, McNaughton P A. Ion channels gated by heat. Proc Natl Acad Sci USA, 1999, 96: 7658–7663

    Article  Google Scholar 

  6. Deng C, Li C M. Endpoint bilinear estimates and applications to the two-dimensional Poisson-Nernst-Planck system. Nonlinearity, 2013, 26: 2993–3009

    Article  MathSciNet  MATH  Google Scholar 

  7. Duan R J, Ruan L Z, Zhu C J. Optimal decay rates to conservation laws with diffusion-type terms of regularity-gain and regularity-loss. Math Models Methods Appl Sci, 2012, 22: 1250012 39 pp

    Article  MathSciNet  MATH  Google Scholar 

  8. Eisenberg R S. Computing the field in proteins and channels. J Membrane Biol, 1996, 150: 1–25

    Article  Google Scholar 

  9. Eisenberg R S. From structure to function in open ionic channels. J Membrane Biol, 1999, 171: 1–24

    Article  Google Scholar 

  10. Elad D, Gavish N. Finite domain effects in steady state solutions of Poisson-Nernst-Planck equations. SIAM J Appl Math, 2019, 79: 1030–1050

    Article  MathSciNet  MATH  Google Scholar 

  11. Eisenberg B, Liu W S. Poisson-Nernst-Planck systems for ion channels with permanent charges. SIAM J Math Anal, 2007, 38: 1932–1966

    Article  MathSciNet  MATH  Google Scholar 

  12. Gajewski H. On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors. Z Angew Math Mech, 1985, 65: 101–108

    Article  MathSciNet  MATH  Google Scholar 

  13. Gagneux G, Millet O. A survey on properties of Nernst-Planck-Poisson system. Application to ionic transport in porous media. Appl Math Model, 2016, 40: 846–858

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo Y, Wang Y J. Decay of dissipative equations and negative Sobolev spaces. Commun Part Diff Equ, 2012, 37: 2165–2208

    Article  MathSciNet  MATH  Google Scholar 

  15. Hsieh C Y. Global existence of solutions for the Poisson-Nernst-Planck system with steric effects. Nonlinear Anal Real World Appl, 2019, 50: 34–54

    Article  MathSciNet  MATH  Google Scholar 

  16. Hsieh C Y, Lin T C. Exponential decay estimates for the stability of boundary layer solutions to poisson-nernst-planck systems: One spatial dimension case. SIAM J Appl Math, 2015, 47: 3442–3465

    Article  MathSciNet  MATH  Google Scholar 

  17. Hsieh C Y, Lin T C, Liu C, Liu P. Global existence of the non-isothermal Poisson-Nernst-Planck-Fourier system. J Differential Equations, 2020, 269: 7287–7310

    Article  MathSciNet  MATH  Google Scholar 

  18. Jerome J W. Analysis of charge transport. A mathematical study of semiconductor devices. Berlin: Springer-Verlag, 1996

    Google Scholar 

  19. Jordan P C, Bacquet R J, McCammon J A, Tran P. How electrolyte shielding influences the electrical potential in transmembrane ion channels. Biophysical Journal, 1989, 55: 1041–1052

    Article  Google Scholar 

  20. Ji L J, Liu P, Xu Z L, Zhou S G. Asymptotic analysis on dielectric boundary effects of modified Poisson-Nernst-Planck equations. SIAM J Appl Math, 2018, 78: 1802–1822

    Article  MathSciNet  MATH  Google Scholar 

  21. Jiang N, Luo Y L, Zhang X. Long time stability of admissible equilibria in Poisson-Nernst-Planck-Fourier system. arXiv:1910.04094

  22. Liu W S. Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems. SIAM J Appl Math, 2005, 65: 754–766

    Article  MathSciNet  MATH  Google Scholar 

  23. Lin T C, Eisenberg B. A new approach to the lennard-jones potential and a new model: Pnp-steric equations. Commun Math Sci, 2014, 12: 149–173

    Article  MathSciNet  MATH  Google Scholar 

  24. Lin T C, Eisenberg B. Multiple solutions of steady-state Poisson-Nernst-Planck equations with steric effects. Nonlinearity, 2015, 28: 2053–2080

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu P, Wu S, Liu C. Non-isothermal electrokinetics: energetic variational approach. Commun Math Sci, 2018, 16: 1451–1463

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu W S, Xu H G. A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow. J Differential Equations, 2015, 258: 1192–1228

    Article  MathSciNet  MATH  Google Scholar 

  27. Mock M S. An initial value problem from semiconductor device theory. SIAM J Math Anal, 1974, 5: 597–612

    Article  MathSciNet  MATH  Google Scholar 

  28. Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67–104

    MathSciNet  MATH  Google Scholar 

  29. Markowich P A, Ringhofer C A, Schmeiser C. Semiconductor equations. Vienna: Springer-Verlag, 1990

    Book  MATH  Google Scholar 

  30. Nirenberg L. On elliptic partial differential equations. Ann Scuola Norm Sup Pisa Cl Sci, 1959, 13: 115–162

    MathSciNet  MATH  Google Scholar 

  31. Nonner W, Chen D P, Eisenberg B. Progress and prospects in permeation. J Gen Physiol, 1999, 113: 773–782

    Article  Google Scholar 

  32. Ogawa T, Shimizu S. The drift-diffusion system in two-dimensional critical Hardy space. J Funct Anal, 2008, 255: 1107–1138

    Article  MathSciNet  MATH  Google Scholar 

  33. Park J H, Jerome J W. Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study. SIAM J Appl Math, 1997, 57: 609–630

    Article  MathSciNet  MATH  Google Scholar 

  34. Promislow K, Stockie J M. Adiabatic relaxation of convective-diffusive gas transport in a porous fuel cell electrode. SIAM J Appl Math, 2001, 62: 180–205

    Article  MathSciNet  MATH  Google Scholar 

  35. Reubish D S, Emerling D E, DeFalco J, Steiger D, Victoria C L, Vincent F. Functional assessment of temperature-gated ion-channel activity using a real-time PCR machine. BioTechniques, 2009, 47: iii–ix. PMID: 19852757

    Article  Google Scholar 

  36. Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series. No 30. Princeton, NJ: Princeton University Press, 1970

    MATH  Google Scholar 

  37. Sohr H. The Navier-Stokes equations. Birkhäuser Advanced Texts: Basler Lehrbücher. Basel: Birkhäuser Verlag, 2001

    Google Scholar 

  38. Song Z L, Cao X L, Huang H X. Electroneutral models for a multidimensional dynamic Poisson-Nernst-Planck system. Phys Rev E, 2018, 98: 032404

    Article  Google Scholar 

  39. Schoch R B, Han J, Renaud P. Transport phenomena in nanofluidics. Rev Mod Phys, 2008, 80: 839

    Article  Google Scholar 

  40. Schuss Z, Nadler B, Eisenberg R S. Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model. Phys Rev E, 2001, 64: 036116

    Article  Google Scholar 

  41. Wu Y S, Tan Z. Asymptotic behavior of the Stokes approximation equations for compressible flows in ℝ3. Acta Mathematica Scientia, 2015, 35B(3): 746–760

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang Y H, Wu G C. Global existence and asymptotic behavior for the 3D compressible non-isentropic Euler equations with damping. Acta Mathematica Scientia, 2014, 34B(2): 424–434

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Leilei Tong  (童雷雷)

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Xiamen, 361005, China

    Zhong Tan  (谭忠)

  3. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Xu Zhang  (张旭)

Authors
  1. Leilei Tong  (童雷雷)
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  2. Zhong Tan  (谭忠)
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  3. Xu Zhang  (张旭)
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Corresponding author

Correspondence to Xu Zhang  (张旭).

Additional information

The work of the first author was supported by the National Natural Science Foundation of China (12001077), the Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202000618) and Chongqing University of Posts and Telecommunications startup fund (A2018-128). The second author was supported by the National Natural Science Foundation of China (11926316, 11531010). The third author was supported by National Natural Science Foundation of China (11901537).

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Tong, L., Tan, Z. & Zhang, X. The Time Decay Rates of the Classical Solution to the Poisson-Nernst-Planck-Fourier Equations in ℝ3. Acta Math Sci 42, 1081–1102 (2022). https://doi.org/10.1007/s10473-022-0315-5

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  • DOI: https://doi.org/10.1007/s10473-022-0315-5

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