International Journal of Differential Equations

Global Existence and Asymptotic Behavior of Self-Similar Solutions for the Navier-Stokes-Nernst-Planck-Poisson System in 3

Jihong Zhao, Chao Deng, and Shangbin Cui

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Abstract

We study the Navier-Stokes-Nernst-Planck-Poisson system modeling the flow of electrohydrodynamics. For small initial data, the global existence, uniqueness, and asymptotic stability as time goes to infinity of self-similar solutions to the Cauchy problem of this system posed in the whole three dimensional space are proved in the function spaces of pseudomeasure type.

Article information

Source
Int. J. Differ. Equ., Volume 2011 (2011), Article ID 329014, 19 pages.

Dates
Received: 5 May 2011
Accepted: 5 September 2011
First available in Project Euclid: 25 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485313251

Digital Object Identifier
doi:10.1155/2011/329014

Mathematical Reviews number (MathSciNet)
MR2847600

Zentralblatt MATH identifier
1234.35204

Citation

Zhao, Jihong; Deng, Chao; Cui, Shangbin. Global Existence and Asymptotic Behavior of Self-Similar Solutions for the Navier-Stokes-Nernst-Planck-Poisson System in ${\Bbb R}^{3}$. Int. J. Differ. Equ. 2011 (2011), Article ID 329014, 19 pages. doi:10.1155/2011/329014. https://projecteuclid.org/euclid.ijde/1485313251


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References

  • I. Rubinstein, Electro-Diffusion of Ions, vol. 11 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1990.
  • J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace,” Acta Mathematica, vol. 63, no. 1, pp. 193–248, 1934.
  • J. Bourgain and N. Pavlović, “Ill-posedness of the Navier-Stokes equations in a critical space in 3D,” Journal of Functional Analysis, vol. 255, no. 9, pp. 2233–2247, 2008.
  • M. Cannone, “Harmonic analysis tools for solving the incompressible Navier-Stokes equations,” in Handbook of Mathematical Fluid Dynamics, S. Friedlander and D. Serre, Eds., vol. 3, pp. 161–244, Elsevier, Amsterdam, The Netherlands, 2004.
  • H. Fujita and T. Kato, “On the Navier-Stokes initial value problem. I,” Archive for Rational Mechanics and Analysis, vol. 16, pp. 269–315, 1964.
  • T. Kato, “Strong ${L}^{p}$-solutions of the Navier-Stokes equation in ${\mathbb{R}}^{m}$, with applications to weak solu-tions,” Mathematische Zeitschrift, vol. 187, no. 4, pp. 471–480, 1984.
  • H. Koch and D. Tataru, “Well-posedness for the Navier-Stokes equations,” Advances in Mathematics, vol. 157, no. 1, pp. 22–35, 2001.
  • P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, vol. 431 of Chapman & Hall/ CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.
  • P. Debye and E. Hückel, “Zur theorie der elektrolyte, II: das grenzgesetz für die elektrische leitfähig-keit,” Physik Zeitschrift, vol. 24, pp. 305–325, 1923.
  • P. Biler, W. Hebisch, and T. Nadzieja, “The Debye system: existence and large time behavior of solu-tions,” Nonlinear Analysis, vol. 23, no. 9, pp. 1189–1209, 1994.
  • P. Biler and J. Dolbeault, “Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems,” Annales Henri Poincaré, vol. 1, no. 3, pp. 461–472, 2000.
  • H. Gajewski and K. Gröger, “On the basic equations for carrier transport in semiconductors,” Journal of Mathematical Analysis and Applications, vol. 113, no. 1, pp. 12–35, 1986.
  • G. Karch, “Scaling in nonlinear parabolic equations,” Journal of Mathematical Analysis and Applications, vol. 234, no. 2, pp. 534–558, 1999.
  • M. Kurokiba and T. Ogawa, “Well-posedness for the drift-diffusion system in ${L}^{p}$ arising from the semiconductor device simulation,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 1052–1067, 2008.
  • T. Ogawa and S. Shimizu, “The drift-diffusion system in two-dimensional critical Hardy space,” Journal of Functional Analysis, vol. 255, no. 5, pp. 1107–1138, 2008.
  • T. Ogawa and M. Yamamoto, “Asymptotic behavior of solutions to drift-diffusion system with generalized dissipation,” Mathematical Models & Methods in Applied Sciences, vol. 19, no. 6, pp. 939–967, 2009.
  • J. W. Jerome, “Analytical approaches to charge transport in a moving medium,” Transport Theory and Statistical Physics, vol. 31, no. 4-6, pp. 333–366, 2002.
  • M. Schmuck, “Analysis of the Navier-Stokes-Nernst-Planck-Poisson system,” Mathematical Models & Methods in Applied Sciences, vol. 19, no. 6, pp. 993–1015, 2009.
  • R. J. Ryham, “Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electrohydrodynamicsčommentComment on ref. [25?]: Please update the information of this reference, if possible.,” submitted to Analysis of PDEs, http://arxiv.org/abs/0910.4973v1.
  • J. Zhao, C. Deng, and S. Cui, “Well-posedness of a dissipative system modeling electrohydrodynamics in Lebesgue spaces,” Differential Equations & Applications, vol. 3, no. 3, pp. 427–448, 2011.
  • M. Longaretti, B. Chini, J. W. Jerome, and R. Sacco, “Electrochemical modeling and characterization of voltage operated channels in nano-bio-electronics,” Sensor Letters, vol. 6, no. 1, pp. 49–56, 2008.
  • M. Longaretti, B. Chini, J. W. Jerome, and R. Sacco, “Computational modeling and simulation of complex systems in bio-electronics,” Journal of Computational Electronics, vol. 7, no. 1, pp. 10–13, 2008.
  • M. Longaretti, G. Marino, B. Chini, J. W. Jerome, and R. Sacco, “Computational models in nano-bioelectronics: simulation of ionic transport in voltage operated channels,” Journal of Nanoscience and Nanotechnology, vol. 8, no. 7, pp. 3686–3694, 2008.
  • G. Tian and Z. Xin, “One-point singular solutions to the Navier-Stokes equations,” Topological Methods in Nonlinear Analysis, vol. 11, no. 1, pp. 135–145, 1998.
  • P. Biler, M. Cannone, I. A. Guerra, and G. Karch, “Global regular and singular solutions for a model of gravitating particles,” Mathematische Annalen, vol. 330, no. 4, pp. 693–708, 2004.
  • G. Benfatto, R. Esposito, and M. Pulvirenti, “Planar Navier-Stokes flow for singular initial data,” Nonlinear Analysis, vol. 9, no. 6, pp. 533–545, 1985.
  • M. Cannone and G. Karch, “Smooth or singular solutions to the Navier-Stokes system?” Journal of Differential Equations, vol. 197, no. 2, pp. 247–274, 2004.
  • Y. Le Jan and A. S. Sznitman, “Stochastic cascades and 3-dimensional Navier-Stokes equations,” Probability Theory and Related Fields, vol. 109, no. 3, pp. 343–366, 1997.
  • E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1970.
  • M. Yamazaki, “The Navier-Stokes equations in the weak-${L}^{n}$ space with time-dependent external force,” Mathematische Annalen, vol. 317, no. 4, pp. 635–675, 2000.