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Diffusion Limit of Kinetic Equations for Multiple Species Charged Particles

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Abstract

In ionic solutions, there are multi-species charged particles (ions) with different properties like mass, charge etc. Macroscopic continuum models like the Poisson–Nernst–Planck (PNP) systems have been extensively used to describe the transport and distribution of ionic species in the solvent. Starting from the kinetic theory for the ion transport, we study a Vlasov–Poisson–Fokker–Planck (VPFP) system in a bounded domain with reflection boundary conditions for charge distributions and prove that the global renormalized solutions of the VPFP system converge to the global weak solutions of the PNP system, as the small parameter related to the scaled thermal velocity and mean free path tends to zero. Our results may justify the PNP system as a macroscopic model for the transport of multi-species ions in dilute solutions.

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Authors and Affiliations

  1. Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Hao Wu

  2. Department of Mathematics, National Taiwan University, No.1, Sec. 4, Roosevelt Road, Taipei 106, Taiwan

    Tai-Chia Lin

  3. Department of Mathematics, Penn State University, State College, PA, 16802, USA

    Chun Liu

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  1. Hao Wu
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  2. Tai-Chia Lin
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  3. Chun Liu
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Correspondence to Hao Wu.

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Communicated by F. Lin

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Wu, H., Lin, TC. & Liu, C. Diffusion Limit of Kinetic Equations for Multiple Species Charged Particles. Arch Rational Mech Anal 215, 419–441 (2015). https://doi.org/10.1007/s00205-014-0784-3

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