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

Martin Burger; Ina Humpert; Jan-Frederik Pietschmann

doi:10.3934/krm.2020009

Article Contents

2020, Volume 13, Issue 2: 249-277. Doi: 10.3934/krm.2020009

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

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
^* Corresponding author: Ina Humpert

Received: November 30, 2018

Revised: September 30, 2019

Published: January 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q84, 35K57, 35Q92.

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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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