May  2016, 36(5): 2915-2930. doi: 10.3934/dcds.2016.36.2915

Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion

1. 

Technische Universität München, Zentrum für Mathematik, Boltzmannstr. 3, 85747 Garching, Germany

Received  March 2015 Revised  September 2015 Published  October 2015

We investigate a Poisson-Nernst-Planck type system in three spatial dimensions where the strength of the electric drift depends on a possibly small parameter and the particles are assumed to diffuse quadratically. On grounds of the global existence result proved by Kinderlehrer, Monsaingeon and Xu (2015) using the formal Wasserstein gradient flow structure of the system, we analyse the long-time behaviour of weak solutions. We prove under the assumption of uniform convexity of the external drift potentials that the system possesses a unique steady state. If the strength of the electric drift is sufficiently small, we show convergence of solutions to the respective steady state at an exponential rate using entropy-dissipation methods.
Citation: Jonathan Zinsl. Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2915-2930. doi: 10.3934/dcds.2016.36.2915
References:
[1]

M. Agueh, Rates of decay to equilibria for p-Laplacian type equations,, Nonlinear Anal., 68 (2008), 1909. doi: 10.1016/j.na.2007.01.043. Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, 2nd edition, (2008). Google Scholar

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A. Arnold, P. Markowich and G. Toscani, On large time asymptotics for drift-diffusion-Poisson systems,, Transport Theory Statist. Phys., 29 (2000), 571. doi: 10.1080/00411450008205893. Google Scholar

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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,, Comm. Partial Differential Equations, 26 (2001), 43. doi: 10.1081/PDE-100002246. Google Scholar

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N. Ben Abdallah, F. Méhats and N. Vauchelet, A note on the long time behavior for the drift-diffusion-Poisson system,, C. R. Math. Acad. Sci. Paris, 339 (2004), 683. doi: 10.1016/j.crma.2004.09.025. Google Scholar

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P. Biler and J. Dolbeault, Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems,, Ann. Henri Poincaré, 1 (2000), 461. doi: 10.1007/s000230050003. Google Scholar

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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, 30 (2001), 521. doi: 10.1081/TT-100105936. Google Scholar

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A. Blanchet, E. A. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model,, J. Funct. Anal., 262 (2012), 2142. doi: 10.1016/j.jfa.2011.12.012. Google Scholar

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A. Blanchet and P. Laurençot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\mathbbR^d,\ d\ge3$,, Comm. Partial Differential Equations, 38 (2013), 658. doi: 10.1080/03605302.2012.757705. Google Scholar

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A. Blanchet, J. A. Carrillo, D. Kinderlehrer, M. Kowalczyk, P. Laurençot and S. Lisini, A hybrid variational principle for the Keller-Segel system in $\mathbbR^2$,, ESAIM: M2AN, (2015), 1. doi: 10.1051/m2an/2015021. Google Scholar

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J. A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity,, Indiana Univ. Math. J., 49 (2000), 113. doi: 10.1512/iumj.2000.49.1756. Google Scholar

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J. A. Carrillo, M. Di Francesco and G. Toscani, Intermediate asymptotics beyond homogeneity and self-similarity: Long time behavior for $u_t=\Delta\phi(u)$,, Arch. Ration. Mech. Anal., 180 (2006), 127. doi: 10.1007/s00205-005-0403-4. Google Scholar

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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. Iberoamericana, 19 (2003), 971. doi: 10.4171/RMI/376. Google Scholar

[14]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media,, Arch. Ration. Mech. Anal., 179 (2006), 217. doi: 10.1007/s00205-005-0386-1. Google Scholar

[15]

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. doi: 10.1007/s00605-008-0532-6. Google Scholar

[16]

U. Gianazza, G. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation,, Arch. Ration. Mech. Anal., 194 (2009), 133. doi: 10.1007/s00205-008-0186-5. Google Scholar

[17]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar

[18]

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (2001). doi: 10.1007/978-3-0348-8334-4. Google Scholar

[19]

D. Kinderlehrer, L. Monsaingeon and X. Xu, A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations,, ESAIM: COCV, (2015), 1. doi: 10.1051/cocv/2015043. Google Scholar

[20]

D. Kinderlehrer and M. Kowalczyk, The Janossy effect and hybrid variational principles,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 153. doi: 10.3934/dcdsb.2009.11.153. Google Scholar

[21]

P. Laurençot and B.-V. Matioc, A gradient flow approach to a thin film approximation of the Muskat problem,, Calc. Var. Partial Differential Equations, 47 (2013), 319. doi: 10.1007/s00526-012-0520-5. Google Scholar

[22]

E. H. Lieb and M. Loss, Analysis,, 2nd edition, (2001). doi: 10.1090/gsm/014. Google Scholar

[23]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

[24]

D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type,, Comm. Partial Differential Equations, 34 (2009), 1352. doi: 10.1080/03605300903296256. Google Scholar

[25]

R. J. McCann, A convexity principle for interacting gases,, Adv. Math., 128 (1997), 153. doi: 10.1006/aima.1997.1634. Google Scholar

[26]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101. doi: 10.1081/PDE-100002243. Google Scholar

[27]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (2003). doi: 10.1007/b12016. Google Scholar

[28]

J. Zinsl, Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure,, Monatsh. Math., 174 (2014), 653. doi: 10.1007/s00605-013-0573-3. Google Scholar

[29]

J. Zinsl, A note on the variational analysis of the parabolic-parabolic Keller-Segel system in one spatial dimension,, C. R. Math. Acad. Sci. Paris, 353 (2015), 849. doi: 10.1016/j.crma.2015.06.014. Google Scholar

[30]

J. Zinsl and D. Matthes, Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis,, Anal. PDE, 8 (2015), 425. doi: 10.2140/apde.2015.8.425. Google Scholar

[31]

J. Zinsl and D. Matthes, Transport distances and geodesic convexity for systems of degenerate diffusion equations,, Calc. Var. Partial Differential Equations, (2015), 1. doi: 10.1007/s00526-015-0909-z. Google Scholar

show all references

References:
[1]

M. Agueh, Rates of decay to equilibria for p-Laplacian type equations,, Nonlinear Anal., 68 (2008), 1909. doi: 10.1016/j.na.2007.01.043. Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, 2nd edition, (2008). Google Scholar

[3]

A. Arnold, P. Markowich and G. Toscani, On large time asymptotics for drift-diffusion-Poisson systems,, Transport Theory Statist. Phys., 29 (2000), 571. doi: 10.1080/00411450008205893. Google Scholar

[4]

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,, Comm. Partial Differential Equations, 26 (2001), 43. doi: 10.1081/PDE-100002246. Google Scholar

[5]

N. Ben Abdallah, F. Méhats and N. Vauchelet, A note on the long time behavior for the drift-diffusion-Poisson system,, C. R. Math. Acad. Sci. Paris, 339 (2004), 683. doi: 10.1016/j.crma.2004.09.025. Google Scholar

[6]

P. Biler and J. Dolbeault, Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems,, Ann. Henri Poincaré, 1 (2000), 461. doi: 10.1007/s000230050003. Google Scholar

[7]

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, 30 (2001), 521. doi: 10.1081/TT-100105936. Google Scholar

[8]

A. Blanchet, E. A. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model,, J. Funct. Anal., 262 (2012), 2142. doi: 10.1016/j.jfa.2011.12.012. Google Scholar

[9]

A. Blanchet and P. Laurençot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\mathbbR^d,\ d\ge3$,, Comm. Partial Differential Equations, 38 (2013), 658. doi: 10.1080/03605302.2012.757705. Google Scholar

[10]

A. Blanchet, J. A. Carrillo, D. Kinderlehrer, M. Kowalczyk, P. Laurençot and S. Lisini, A hybrid variational principle for the Keller-Segel system in $\mathbbR^2$,, ESAIM: M2AN, (2015), 1. doi: 10.1051/m2an/2015021. Google Scholar

[11]

J. A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity,, Indiana Univ. Math. J., 49 (2000), 113. doi: 10.1512/iumj.2000.49.1756. Google Scholar

[12]

J. A. Carrillo, M. Di Francesco and G. Toscani, Intermediate asymptotics beyond homogeneity and self-similarity: Long time behavior for $u_t=\Delta\phi(u)$,, Arch. Ration. Mech. Anal., 180 (2006), 127. doi: 10.1007/s00205-005-0403-4. Google Scholar

[13]

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. Iberoamericana, 19 (2003), 971. doi: 10.4171/RMI/376. Google Scholar

[14]

J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media,, Arch. Ration. Mech. Anal., 179 (2006), 217. doi: 10.1007/s00205-005-0386-1. Google Scholar

[15]

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. doi: 10.1007/s00605-008-0532-6. Google Scholar

[16]

U. Gianazza, G. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation,, Arch. Ration. Mech. Anal., 194 (2009), 133. doi: 10.1007/s00205-008-0186-5. Google Scholar

[17]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. Google Scholar

[18]

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (2001). doi: 10.1007/978-3-0348-8334-4. Google Scholar

[19]

D. Kinderlehrer, L. Monsaingeon and X. Xu, A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations,, ESAIM: COCV, (2015), 1. doi: 10.1051/cocv/2015043. Google Scholar

[20]

D. Kinderlehrer and M. Kowalczyk, The Janossy effect and hybrid variational principles,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 153. doi: 10.3934/dcdsb.2009.11.153. Google Scholar

[21]

P. Laurençot and B.-V. Matioc, A gradient flow approach to a thin film approximation of the Muskat problem,, Calc. Var. Partial Differential Equations, 47 (2013), 319. doi: 10.1007/s00526-012-0520-5. Google Scholar

[22]

E. H. Lieb and M. Loss, Analysis,, 2nd edition, (2001). doi: 10.1090/gsm/014. Google Scholar

[23]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

[24]

D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type,, Comm. Partial Differential Equations, 34 (2009), 1352. doi: 10.1080/03605300903296256. Google Scholar

[25]

R. J. McCann, A convexity principle for interacting gases,, Adv. Math., 128 (1997), 153. doi: 10.1006/aima.1997.1634. Google Scholar

[26]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101. doi: 10.1081/PDE-100002243. Google Scholar

[27]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (2003). doi: 10.1007/b12016. Google Scholar

[28]

J. Zinsl, Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure,, Monatsh. Math., 174 (2014), 653. doi: 10.1007/s00605-013-0573-3. Google Scholar

[29]

J. Zinsl, A note on the variational analysis of the parabolic-parabolic Keller-Segel system in one spatial dimension,, C. R. Math. Acad. Sci. Paris, 353 (2015), 849. doi: 10.1016/j.crma.2015.06.014. Google Scholar

[30]

J. Zinsl and D. Matthes, Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis,, Anal. PDE, 8 (2015), 425. doi: 10.2140/apde.2015.8.425. Google Scholar

[31]

J. Zinsl and D. Matthes, Transport distances and geodesic convexity for systems of degenerate diffusion equations,, Calc. Var. Partial Differential Equations, (2015), 1. doi: 10.1007/s00526-015-0909-z. Google Scholar

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