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April  2020, 13(2): 249-277. doi: 10.3934/krm.2020009

On Fokker-Planck equations with In- and Outflow of Mass

Martin Burger 1, , Ina Humpert 2,, and  Jan-Frederik Pietschmann 3,

1. 

Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik, Cauerstr. 11, 91058 Erlangen, Germany
2. 

Westfälische Wilhelms-Universität Münster, Institut für Angewandte Mathematik: Analysis und Numerik, Orleans-Ring 10, 48149 Münster, Germany
3. 

Technische Universität Chemnitz, Fakultät für Mathematik, Reichenhainer Str. 41, 09126 Chemnitz, Germany

* Corresponding author: Ina Humpert

Received  December 2018 Revised  October 2019 Published  January 2020

Motivated by modeling transport processes in the growth of neurons, we present results on (nonlinear) Fokker-Planck equations where the total mass is not conserved. This is either due to in- and outflow boundary conditions or to spatially distributed reaction terms. We are able to prove exponential decay towards equilibrium using entropy methods in several situations. As there is no conservation of mass it is difficult to exploit the gradient flow structure of the differential operator which renders the analysis more challenging. In particular, classical logarithmic Sobolev inequalities are not applicable any more. Our analytic results are illustrated by extensive numerical studies.

Keywords: Fokker-Planck equations, Entropy methods, Exponential decay, Mass evolution, Logarithmic-Sobolev inequality.
Mathematics Subject Classification: Primary: 35Q84, 35K57, 35Q92.
Citation: Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009
References:
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M. Burger and J.-F. Pietschmann, Flow characteristics in a crowded transport model, Nonlinearity, 29 (2016), 3528-3550.  doi: 10.1088/0951-7715/29/11/3528.  Google Scholar

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[13]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019.  Google Scholar

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W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Annals of Mathematics, 55 (1952), 468-519.  doi: 10.2307/1969644.  Google Scholar

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K. Fellner and M. Kniely, Uniform convergence to equilibrium for a family of drift-diffusion models with trap-assisted recombination and the limiting shockley–read–hall model, arXiv preprint, arXiv: 1703.02881, 2017. Google Scholar

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K. Fellner, L. Neumann and C. Schmeiser, Convergence to global equilibrium for spatially inhomogeneous kinetic models of non-micro-reversible processes, Monatshefte Für Mathematik, 141 (2004), 289-299.  doi: 10.1007/s00605-002-0058-2.  Google Scholar

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S. N. Gomes, A. M. Stuart and M.-T. Wolfram, Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM J. Appl. Math., 79 (2019), 1475–1500, arXiv1809.08046. doi: 10.1137/18M1215980.  Google Scholar

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J. Haskovec, S. Hittmeir, P. Markowich and A. Mielke, Decay to equilibrium for energy-reaction-diffusion systems, SIAM J. Math. Anal, 50 (2018), 1037-1075.  doi: 10.1137/16M1062065.  Google Scholar

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R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the fokker–planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[21]

D. Le and H. Smith, Strong positivity of solutions to parabolic and elliptic equations on nonsmooth domains, Journal of Mathematical Analysis and Applications, 275 (2002), 208-221.  doi: 10.1016/S0022-247X(02)00314-1.  Google Scholar

[22]

M. Liero and A. Mielke, Gradient structures and geodesic convexity for reaction–diffusion systems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120346, 28pp. doi: 10.1098/rsta.2012.0346.  Google Scholar

[23]

P. A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis, Mat. Contemp., 19 (2000), 1–29. Ⅵ Workshop on Partial Differential Equations, Part Ⅱ (Rio de Janeiro, 1999).  Google Scholar

[24]

P. K. Mattila and P. Lappalainen, Filopodia: Molecular architecture and cellular functions, Nature Reviews Molecular Cell Biology, 9 (2008), 446-454.  doi: 10.1038/nrm2406.  Google Scholar

[25]

A. Mielke, A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.  doi: 10.1088/0951-7715/24/4/016.  Google Scholar

[26]

T. Namba, Y. Funahashi, S. Nakamuta, C. Xu, T. Takano and K. Kaibuchi, Extracellular and intracellular signaling for neuronal polarity, Physiological Reviews, 95 (2015), 995-1024.  doi: 10.1152/physrev.00025.2014.  Google Scholar

[27]

I. Naoyuki, T. Michinori and S. Yuichi, Systems biology of symmetry breaking during neuronal polarity formation, Developmental Neurobiology, 71 584–593. Google Scholar

[28]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Communications in Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.  Google Scholar

[29]

D. Roland, On poincaré, friedrichs and korns inequalities on domains and hypersurfaces, arXiv preprint, arXiv: 1504.01677, 2015. Google Scholar

[30]

T. Takano, C. Xu, Y. Funahashi, T. Namba and K. Kaibuchi, Neuronal polarization, Development, 142 (2015), 2088-2093.  doi: 10.1242/dev.114454.  Google Scholar

show all references

References:
[1]

A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of many-particle systems: An essay on recent research, Monatsh. Math, 142 (2004), 35-43.  doi: 10.1007/s00605-004-0239-2.  Google Scholar

[2]

M. Burger, M. Di Francesco and Y. Dolak-Struss, The keller–segel model for chemotaxis with prevention of overcrowding: Linear vs. nonlinear diffusion, SIAM Journal on Mathematical Analysis, 38 (2006), 1288-1315.  doi: 10.1137/050637923.  Google Scholar

[3]

M. Burger and J.-F. Pietschmann, Flow characteristics in a crowded transport model, Nonlinearity, 29 (2016), 3528-3550.  doi: 10.1088/0951-7715/29/11/3528.  Google Scholar

[4]

J. Carrillo, P. Laurenccot and J. Rosado, Fermi–Dirac–Fokker–Planck equation: Well-posedness & long-time asymptotics, Journal of Differential Equations, 247 (2009), 2209-2234.  doi: 10.1016/j.jde.2009.07.018.  Google Scholar

[5]

J. A. Carrillo and J. L. Vázquez, Fine asymptotics for fast diffusion equations, Communications in Partial Differential Equations, 28 (2003), 1023-1056.  doi: 10.1081/PDE-120021185.  Google Scholar

[6]

M. Del Pino and J. Dolbeault, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, Journal de Mathématiques Pures et Appliquées, 81 (2002), 847-875.  doi: 10.1016/S0021-7824(02)01266-7.  Google Scholar

[7]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Inventiones Mathematicae, 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[8]

L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl, 319 (2006), 157-176.  doi: 10.1016/j.jmaa.2005.07.003.  Google Scholar

[9]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Communications on Pure and Applied Mathematics, 54 (2001), 1-42.  doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[10]

M. Dreher and A. Jüngel, Compact families of piecewise constant functions in lp (0, t; b), Nonlinear Analysis, Theory, Methods and Applications, 75 (2012), 3072-3077.  doi: 10.1016/j.na.2011.12.004.  Google Scholar

[11]

J. Droniou and J.-L. Vázquez, Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions, Calc. Var. Partial Differential Equations, 34 (2009), 413-434.  doi: 10.1007/s00526-008-0189-y.  Google Scholar

[12]

C. M. Elliott and H. Garcke, On the cahn–hilliard equation with degenerate mobility, Siam Journal on Mathematical Analysis, 27 (1996), 404-423.  doi: 10.1137/S0036141094267662.  Google Scholar

[13]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019.  Google Scholar

[14]

W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Annals of Mathematics, 55 (1952), 468-519.  doi: 10.2307/1969644.  Google Scholar

[15]

K. Fellner and M. Kniely, Uniform convergence to equilibrium for a family of drift-diffusion models with trap-assisted recombination and the limiting shockley–read–hall model, arXiv preprint, arXiv: 1703.02881, 2017. Google Scholar

[16]

K. Fellner, L. Neumann and C. Schmeiser, Convergence to global equilibrium for spatially inhomogeneous kinetic models of non-micro-reversible processes, Monatshefte Für Mathematik, 141 (2004), 289-299.  doi: 10.1007/s00605-002-0058-2.  Google Scholar

[17]

S. N. Gomes, A. M. Stuart and M.-T. Wolfram, Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM J. Appl. Math., 79 (2019), 1475–1500, arXiv1809.08046. doi: 10.1137/18M1215980.  Google Scholar

[18]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, London, Melbourne, 1985.  Google Scholar

[19]

J. Haskovec, S. Hittmeir, P. Markowich and A. Mielke, Decay to equilibrium for energy-reaction-diffusion systems, SIAM J. Math. Anal, 50 (2018), 1037-1075.  doi: 10.1137/16M1062065.  Google Scholar

[20]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the fokker–planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[21]

D. Le and H. Smith, Strong positivity of solutions to parabolic and elliptic equations on nonsmooth domains, Journal of Mathematical Analysis and Applications, 275 (2002), 208-221.  doi: 10.1016/S0022-247X(02)00314-1.  Google Scholar

[22]

M. Liero and A. Mielke, Gradient structures and geodesic convexity for reaction–diffusion systems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120346, 28pp. doi: 10.1098/rsta.2012.0346.  Google Scholar

[23]

P. A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis, Mat. Contemp., 19 (2000), 1–29. Ⅵ Workshop on Partial Differential Equations, Part Ⅱ (Rio de Janeiro, 1999).  Google Scholar

[24]

P. K. Mattila and P. Lappalainen, Filopodia: Molecular architecture and cellular functions, Nature Reviews Molecular Cell Biology, 9 (2008), 446-454.  doi: 10.1038/nrm2406.  Google Scholar

[25]

A. Mielke, A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.  doi: 10.1088/0951-7715/24/4/016.  Google Scholar

[26]

T. Namba, Y. Funahashi, S. Nakamuta, C. Xu, T. Takano and K. Kaibuchi, Extracellular and intracellular signaling for neuronal polarity, Physiological Reviews, 95 (2015), 995-1024.  doi: 10.1152/physrev.00025.2014.  Google Scholar

[27]

I. Naoyuki, T. Michinori and S. Yuichi, Systems biology of symmetry breaking during neuronal polarity formation, Developmental Neurobiology, 71 584–593. Google Scholar

[28]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Communications in Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.  Google Scholar

[29]

D. Roland, On poincaré, friedrichs and korns inequalities on domains and hypersurfaces, arXiv preprint, arXiv: 1504.01677, 2015. Google Scholar

[30]

T. Takano, C. Xu, Y. Funahashi, T. Namba and K. Kaibuchi, Neuronal polarization, Development, 142 (2015), 2088-2093.  doi: 10.1242/dev.114454.  Google Scholar

Figure 1.  Sketch of a neuron (1) cell nucleus, 2) dendrite and 3) axon)
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Figure 2.  Sketch of the geometry of boundary in- and outflux in 2D with a possible density
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Figure 3.  Evolution over Time of the Particle Concentration: The time evolution of $ \rho $ solving (3) in comparison to the calculated stationary solution (21) for $ \alpha = 1, \beta = 0.9 $ and initial particle concentration $ \rho(x) = -0.1x +1.2 $. (a) The initial concentration at $ t = 0 $, (b) strong influence of the boundary terms at $ t = 0.05 $, (c) strong influence of the drift term and the diffusion at $ t = 1.5 $, (d) equilibrium state at $ t = 9 $
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Figure 4.  Relative Entropy: The relative entropy (15) for the one dimensional version of (3) for the initial particle concentration $ \rho(x) = -0.1x +1.2 $. (a) Natural logarithm of the relative entropy for $ \alpha = 1, \; \beta = 1 $ and variable scaling factor $ \gamma $ of the potential term, (b) relation between the scaling factor $ \gamma $ and the corresponding slope of the logarithm of the relative entropy
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Figure 5.  Mass Evolution: Two examples for non monotone mass evolution for $ \alpha = 1 $ and $ \beta = 0.9 $
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Figure 6.  Sketch of the geometry of uniform spatial in- and outflux in 2D
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Figure 7.  Evolution over Time of the Particle Concentration: Solution of (26) in comparison to the calculated stationary solution $ \frac{\alpha}{\beta}e^{V(x)} $ for $ \alpha = 1, \beta = 0.9 $, initial particle concentration $ \rho(x) = -0.1x+1.2 $. (a) The initial particle concentration at $ t = 0 $, (b) + (c) particle concentration at $ t = 0.05 $ and $ t = 1.5 $, (d) equilibrium state at $ t = 20 $
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Figure 8.  Relative Entropy: Natural logarithm of the relative entropy(30) for the one dimensional version of (26) for $ \alpha = 1, \beta = 0.9 $, initial concentration $ \rho(x) = 0.1x +1 $
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Figure 9.  Evolution over Time of the Particle Concentration: $ \rho(x,t) $ solving (35) in comparison to the calculated solution (44) for $ \alpha = 1, \beta = 0.9 $, initial particle concentration $ \rho_0(x) = -(x-0.5)^2+1 $. (a) The initial concentration in comparison with the calculated stationary solution at $ t = 0 $, (b) diffusion strongly visible $ t = 0.05 $, (c) transport term strongly visible $ t = 0.35 $, (d) equilibrium state at $ t = 3.7 $
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Figure 10.  Relative Entropy: The logarithm of the relative entropy for the one dimensional version of (35) for $ \alpha = 1, \beta = 0.9 $ and $ \rho_0(x) = -(x-0.5)^2+1 $
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