A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations

Kinderlehrer, David; Monsaingeon, Léonard; Xu, Xiang

doi:10.1051/cocv/2015043

Kinderlehrer, David ¹ ; Monsaingeon, Léonard ² ; Xu, Xiang ³

¹ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
² CAMGSD Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal.
³ Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 137-164.

Résumé

The Poisson−Nernst−Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of global weak solutions in a unified framework for the cases of both linear and nonlinear diffusion. The proof of the main results relies on the derivation of additional estimates based on the flow interchange technique developed by Matthes et al. in [D. Matthes, R.J. McCann and G. Savaré, Commun. Partial Differ. Equ. 34 (2009) 1352–1397].

Reçu le : 2015-01-22
Accepté le : 2015-08-06

MR Zbl | 3 citations dans Numdam

DOI : 10.1051/cocv/2015043

Classification : 35K65, 35K40, 47J30, 35Q92, 35B33
Mots-clés : Optimal transport, systems of parabolic PDEs, nonlocal equations

Affiliations des auteurs :

Kinderlehrer, David ¹ ; Monsaingeon, Léonard ² ; Xu, Xiang ³

@article{COCV_2017__23_1_137_0,
     author = {Kinderlehrer, David and Monsaingeon, L\'eonard and Xu, Xiang},
     title = {A {Wasserstein} gradient flow approach to {Poisson\ensuremath{-}Nernst\ensuremath{-}Planck} equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {137--164},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {1},
     year = {2017},
     doi = {10.1051/cocv/2015043},
     mrnumber = {3601019},
     zbl = {1372.35167},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2015043/}
}

TY  - JOUR
AU  - Kinderlehrer, David
AU  - Monsaingeon, Léonard
AU  - Xu, Xiang
TI  - A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2017
SP  - 137
EP  - 164
VL  - 23
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2015043/
DO  - 10.1051/cocv/2015043
LA  - en
ID  - COCV_2017__23_1_137_0
ER  -

%0 Journal Article
%A Kinderlehrer, David
%A Monsaingeon, Léonard
%A Xu, Xiang
%T A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2017
%P 137-164
%V 23
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2015043/
%R 10.1051/cocv/2015043
%G en
%F COCV_2017__23_1_137_0

Kinderlehrer, David; Monsaingeon, Léonard; Xu, Xiang. A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 137-164. doi : 10.1051/cocv/2015043. http://archive.numdam.org/articles/10.1051/cocv/2015043/

Bibliographie
Cité par

L. Ambrosio and N. Gigli, A user’s guide to optimal transport. In Modelling and optimisation of flows on networks. Vol. 2062 of Lect. Notes Math. Springer, Heidelberg (2013) 1–155. | MR

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. 2nd edition. Birkhäuser Verlag, Basel (2008). | MR | Zbl

A. Arnold, P. Markowich and G. Toscani, On large time asymptotics for drift-diffusion-Poisson systems. Vol. 29 of In Proc. of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998) (2000) 571–581. | MR | Zbl

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Partial Differ. Eq. 26 (2001) 43–100. | DOI | MR | Zbl

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge−Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375–393. | DOI | MR | Zbl

P. Biler and J. Dolbeault, Long time behavior of solutions of Nernst−Planck and Debye−Hückel drift-diffusion systems. Ann. Henri Poincaré (2000) 461–472. | MR | Zbl

P. Biler, J. Dolbeault and P.A. Markowich, Large time asymptotics of nonlinear drift-diffusion systems with Poisson coupling. The Sixteenth International Conference on Transport Theory, Part II (Atlanta, GA 1999). Transport Theory Statist. Phys. 30 (2001) 521–536. | DOI | MR | Zbl

A. Blanchet and Ph. Laurençot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $R^{d}, d \geq 3$ . Commun. Partial Differ. Eq. 38 (2013) 658–686. | DOI | MR | Zbl

A. Blanchet, V. Calvez and J.A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model. SIAM J. Numer. Anal. 46 (2008) 691–721. | DOI | MR | Zbl

A. Blanchet, J.A. Carrillo, D. Kinderlehrer and M. Kowalczyk, Philippe Laurençot and Stefano Lisini. A hybrid variational principle for the Keller−segel system in $R^{2}$ . Preprint [math.AP] (2014). | arXiv | Numdam | MR

J.A. Carrillo, R.J. Mccann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamer. 19 (2003) 971–1018. | DOI | MR | Zbl

M. Di Francesco and M. Wunsch, Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models. Monatsh. Math. 154 (2008) 39–50. | DOI | MR | Zbl

W. Fang and K. Ito, On the time-dependent drift-diffusion model for semiconductors. J. Differ. Eq. 117 (1995) 245–280. | DOI | MR | Zbl

H. Gajewski, On the uniqueness of solutions to the drift-diffusion model of semiconductor devices. Math. Models Methods Appl. Sci. 4 (1994) 121–133. | DOI | MR | Zbl

S. Hastings, D. Kinderlehrer and J.B. Mcleod, Diffusion mediated transport in multiple state systems. SIAM J. Math. Anal. 39 (2007/08) 1208–1230. | DOI | MR | Zbl

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1–17. | DOI | MR | Zbl

A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in electrophoretic and semiconductor modeling. Math. Nachr. 185 (1997) 85–110. | DOI | MR | Zbl

A. Jüngel, Quasi-hydrodynamic semiconductor equations. Vol. 41 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Verlag, Basel (2001). | MR | Zbl

C.E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems. Vol. 83 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1994). | MR | Zbl

D. Kinderlehrer and M. Kowalczyk, The Janossy effect and hybrid variational principles. Discrete Contin. Dyn. Syst. Ser. B 11 (2009) 153–176. | MR | Zbl

M. Kurokiba and T. Ogawa, Well-posedness for the drift-diffusion system in $L^{p}$ arising from the semiconductor device simulation. J. Math. Anal. Appl. 342 (2008) 1052–1067. | DOI | MR | Zbl

Ph. Laurençot and B.-V. Matioc, A gradient flow approach to a thin film approximation of the Muskat problem. Calc. Var. Partial Differ. Eq. 47 319–341, 2013. | DOI | MR | Zbl

El.H. Lieb and M. Loss, Analysis. Vol. 14 of Grad. Stud. Math. American Mathematical Society, Providence, RI, 2nd edition (2001). | MR | Zbl

P.A. Markowich, C.A. Ringhofer and C. Schmeiser, Semiconductor equations. Springer-Verlag, Vienna (1990). | MR | Zbl

D. Matthes and J. Zinsl, Exponential convergence to equilibrium in a coupled gradient flow system modelling chemotaxis. Preprint [math.AP] (2014). | arXiv | MR

D. Matthes, R.J. Mccann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type. Commun. Partial Differ. Equ. 34 (2009) 1352–1397. | DOI | MR | Zbl

R.J. Mccann, A convexity principle for interacting gases. Adv. Math. 128 (1997) 153–179. | DOI | MR | Zbl

A. Mielke. A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24 (2011) 1329–1346. | DOI | MR | Zbl

Felix Otto. Dynamics of labyrinthine pattern formation in magnetic fluids: a mean-field theory. Arch. Rational Mech. Anal. 141 (1998) 63–103. | DOI | MR | Zbl

F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Eq. 26 (2001) 101–174. | DOI | MR | Zbl

M. Schmuck, New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials. J. Math. Phys. 54 (2013) 021504. | DOI | MR | Zbl

J. Simon, Compact sets in the space $L^{p} (0, T; B)$ . Ann. Mat. Pura Appl. 146 (1987) 65–96. | DOI | MR | Zbl

E.M. Stein, Singular integrals and differentiability properties of functions. Princeton Mathematical Series. Princeton University Press, Princeton, N.J. (1970) 30. | MR | Zbl

J.L. Vázquez. The porous medium equation. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007). Mathematical theory. | MR | Zbl

C. Villani, Topics in optimal transportation. Vol. 58 of Grad. Stud. MathAmerican Mathematical Society, Providence, RI (2003). | MR | Zbl

Sh. Xu, P. Sheng and Ch. Liu, An energetic variational approach for ion transport. Commun. Math. Sci. 12 (2014) 779–789. | DOI | MR | Zbl

J. Zinsl, Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure. Monatsh. Math. 174 (2014) 653–679. | DOI | MR | Zbl

Cité par Sources :