Communications in Mathematical Sciences

Volume 12 (2014)

Number 4

An energetic variational approach for ion transport

Pages: 779 – 789

(Fast Communication)



Shixin Xu (Department of Mathematics, University of Science and Technology of China, Hefei, China)

Ping Sheng (Department of Physics, Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong, China)

Chun Liu (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)


The transport and distribution of charged particles are crucial in the study of many physical and biological problems. In this paper, we employ an Energy Variational Approach to derive the coupled Poisson-Nernst-Planck-Navier-Stokes system. All of the physics is included in the choices of corresponding energy law and kinematic transport of particles. The variational derivations give the coupled force balance equations in a unique and deterministic fashion. We also discuss the situations with different types of boundary conditions. Finally, we show that the Onsager’s relation holds for the electrokinetics, near the initial time of a step function applied field.


energetic variational approach, Poisson-Nernst-Planck (PNP) system, (least) action principle, (maximum) dissipation principle, Onsager’s relation

2010 Mathematics Subject Classification

35Q35, 35Q92, 76W05, 92B05

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