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January  2021, 14(1): 321-330. doi: 10.3934/dcdss.2020326

Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system

Karoline Disser

FB Mathematik, TU Darmstadt, Schlossgartenstr. 7, 64293 Darmstadt, Germany

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  April 2019 Revised  November 2019 Published  January 2021 Early access  April 2020

We prove a global existence, uniqueness and regularity result for a two-species reaction-diffusion volume-surface system that includes nonlinear bulk diffusion and nonlinear (weak) cross diffusion on the active surface. A key feature is a proof of upper $ L^{\infty} $-bounds that exploits the entropic gradient structure of the system.

Keywords: Volume-surface system, reaction-diffusion system, $ L^{\infty} $-estimates, entropy method, nonlinear diffusion.
Mathematics Subject Classification: Primary: 35K61, 35K57; Secondary: 35B45, 35A01.
Citation: Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326
References:
[1]

D. Bothe, On the multi-physics of mass-transfer across fluid interfaces, arXiv: 1501.05610.

[2]

D. Bothe, M. Köhne, S. Maier and J. Saal, Global strong solutions for a class of heterogeneous catalysis models, J. Math. Anal. Appl., 445 (2017), 677-709.  doi: 10.1016/j.jmaa.2016.08.016.

[3]

H. Brézis, Opérateurs Maximaux Montones et Semi-groupes de Contractions Dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973.

[4]

K. Disser, Well-posedness for coupled bulk-interface diffusion with mixed boundary conditions, Analysis, 35 (2015), 309-317.  doi: 10.1515/anly-2014-1308.

[5]

K. Disser, Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems, arXiv: 1703.07616, J. Differential Equations, accepted for publication (2020).

[6]

K. Disser, M. Meyries and J. Rehberg, A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces, J. Math. Anal. Appl., 430 (2015), 1102-1123.  doi: 10.1016/j.jmaa.2015.05.041.

[7]

K. Fellner, E. Latos and B. Q. Tang, Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 643-673.  doi: 10.1016/j.anihpc.2017.07.002.

[8]

J. R. Fernández, P. Kalita, S. Migórski, M. C. Muñiz and C. Nuñéz, Existence and uniqueness results for a kinetic model in bulk-surface surfactant dynamics, SIAM J. Math. Anal., 48 (2016), 3065-3089.  doi: 10.1137/15M1012785.

[9]

J. Fischer, Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations, Nonlinear Anal., 159 (2017), 181-207.  doi: 10.1016/j.na.2017.03.001.

[10]

A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM J. Math. Anal., 44 (2012), 3874-3900.  doi: 10.1137/110858847.

[11]

A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. Angew. Math. Phys., 64 (2013), 29-52.  doi: 10.1007/s00033-012-0207-y.

[12]

A. J{ü}ngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.  doi: 10.1088/0951-7715/28/6/1963.

[13]

F. Keil, Complexities in modeling of heterogeneous catalytic reactions, Comput. Math. Appl., 65 (2013), 1674-1697.  doi: 10.1016/j.camwa.2012.11.023.

[14]

S. Kjelstrup and D. Bedeaux, Non-equilibrium Thermodynamics of Heterogeneous Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812779144.

[15]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk- interface interactions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 479-499.  doi: 10.3934/dcdss.2013.6.479.

[16]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.

show all references

References:
[1]

D. Bothe, On the multi-physics of mass-transfer across fluid interfaces, arXiv: 1501.05610.

[2]

D. Bothe, M. Köhne, S. Maier and J. Saal, Global strong solutions for a class of heterogeneous catalysis models, J. Math. Anal. Appl., 445 (2017), 677-709.  doi: 10.1016/j.jmaa.2016.08.016.

[3]

H. Brézis, Opérateurs Maximaux Montones et Semi-groupes de Contractions Dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973.

[4]

K. Disser, Well-posedness for coupled bulk-interface diffusion with mixed boundary conditions, Analysis, 35 (2015), 309-317.  doi: 10.1515/anly-2014-1308.

[5]

K. Disser, Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems, arXiv: 1703.07616, J. Differential Equations, accepted for publication (2020).

[6]

K. Disser, M. Meyries and J. Rehberg, A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces, J. Math. Anal. Appl., 430 (2015), 1102-1123.  doi: 10.1016/j.jmaa.2015.05.041.

[7]

K. Fellner, E. Latos and B. Q. Tang, Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 643-673.  doi: 10.1016/j.anihpc.2017.07.002.

[8]

J. R. Fernández, P. Kalita, S. Migórski, M. C. Muñiz and C. Nuñéz, Existence and uniqueness results for a kinetic model in bulk-surface surfactant dynamics, SIAM J. Math. Anal., 48 (2016), 3065-3089.  doi: 10.1137/15M1012785.

[9]

J. Fischer, Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations, Nonlinear Anal., 159 (2017), 181-207.  doi: 10.1016/j.na.2017.03.001.

[10]

A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM J. Math. Anal., 44 (2012), 3874-3900.  doi: 10.1137/110858847.

[11]

A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. Angew. Math. Phys., 64 (2013), 29-52.  doi: 10.1007/s00033-012-0207-y.

[12]

A. J{ü}ngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.  doi: 10.1088/0951-7715/28/6/1963.

[13]

F. Keil, Complexities in modeling of heterogeneous catalytic reactions, Comput. Math. Appl., 65 (2013), 1674-1697.  doi: 10.1016/j.camwa.2012.11.023.

[14]

S. Kjelstrup and D. Bedeaux, Non-equilibrium Thermodynamics of Heterogeneous Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812779144.

[15]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk- interface interactions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 479-499.  doi: 10.3934/dcdss.2013.6.479.

[16]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.

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