Methods and Applications of Analysis

Volume 13 (2006)

Number 3

On the Well-posedness of a Mathematical Model for Lithium-Ion Battery Systems

Pages: 275 – 298



Jinbiao Wu

Jinchao Xu

Henghui Zou


In this paper, we shall establish the well-posedness of a mathematical model for a special class of electrochemical power device - lithium-ion battery. The underlying partial differential equations in the model involve a (mix and fully) coupled system of quasi-linear elliptic and parabolic equations. By exploring some special structure, we are able to adopt the well-known Nash-Moser- DeGiorgi boot strap to establish suitable a priori supremum estimates for the electric potentials. Using the supremum estimates, we apply the Leray-Schauder theory to establish the existence and uniqueness of a subsystem of elliptic equations that describe the electric potentials in the model. We then employ a Schauder fix point theorem to obtain the local (in time) existence for the whole model. We also consider the global existence of a modified 1-d governing system under additional assumptions. In particular, we are able to derive uniform a priori estimates depending only on the existence time $T$, including the supremum estimates for electric potentials and growth and decay estimates for the concentration $c$. Using the uniform estimates, we prove that the modified system has a solution for all time $t>0$.


Lithium-ion battery; Nash-Moser-DeGiorgi boot strap; fixed point theorems; a priori estimates; Newton-Krylov-multigrid method

2010 Mathematics Subject Classification

35G25, 35G30, 35J70, 35K15, 35K20, 35M10

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