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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 & 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. Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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