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Numerical approximation of von Kármán viscoelastic plates
Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system
FB Mathematik, TU Darmstadt, Schlossgartenstr. 7, 64293 Darmstadt, Germany |
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.
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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