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.
Citation: |
[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.![]() ![]() ![]() |