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Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system
Existence of weak solutions for a sharp interface model for phase separation on biological membranes
Faculty of Mathematics, University of Regensburg, 93040 Regensburg, Germany |
We prove existence of weak solutions of a Mullins-Sekerka equation on a surface that is coupled to diffusion equations in a bulk domain and on the boundary. This model arises as a sharp interface limit of a phase field model to describe the formation of liqid rafts on a cell membrane. The solutions are constructed with the aid of an implicit time discretization and tools from geometric measure theory to pass to the limit.
References:
| [1] |
H. Abels and J. Kampmann, On the sharp interface limit of a model for phase separation on biological membranes, Preprint, arXiv: 1811.12489, 2018. |
| [2] |
H. Abels and M. Röger,
Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2403-2424.
doi: 10.1016/j.anihpc.2009.06.002. |
| [3] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity
Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press,
New York, 2000. |
| [4] |
R. E. Edwards, Functional Analysis, Dover Publications Inc. New York, 1995. |
| [5] |
H. Garcke, J. Kampmann, A. Rätz and M. Röger,
A coupled surface-Cahn-Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci., 26 (2016), 1149-1189.
doi: 10.1142/S0218202516500275. |
| [6] |
S. Luckhaus, The Stefan problem with the Gibbs-Thomson law, Preprint Univ. Pisa, 591 (1991). |
| [7] |
S. Luckhaus and T. Sturzenhecker,
Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), 253-271.
doi: 10.1007/BF01205007. |
| [8] |
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, volume 135 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2012. An
introduction to geometric measure theory.
doi: 10.1017/CBO9781139108133. |
| [9] |
M. Röger,
Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation, Interfaces Free Bound., 6 (2004), 105-133.
doi: 10.4171/IFB/93. |
| [10] |
R. Schätzle,
Hypersurfaces with mean curvature given by an ambient {S}obolev function, J. Differential Geom., 58 (2001), 371-420.
doi: 10.4310/jdg/1090348353. |
| [11] |
J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987),
65–96.
doi: 10.1007/BF01762360. |
| [12] |
L. Simon, Lectures on Geometric Measure Theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University., Australian National University Centre for Mathematical Analysis, Canberra, 1983. |
show all references
References:
| [1] |
H. Abels and J. Kampmann, On the sharp interface limit of a model for phase separation on biological membranes, Preprint, arXiv: 1811.12489, 2018. |
| [2] |
H. Abels and M. Röger,
Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2403-2424.
doi: 10.1016/j.anihpc.2009.06.002. |
| [3] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity
Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press,
New York, 2000. |
| [4] |
R. E. Edwards, Functional Analysis, Dover Publications Inc. New York, 1995. |
| [5] |
H. Garcke, J. Kampmann, A. Rätz and M. Röger,
A coupled surface-Cahn-Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci., 26 (2016), 1149-1189.
doi: 10.1142/S0218202516500275. |
| [6] |
S. Luckhaus, The Stefan problem with the Gibbs-Thomson law, Preprint Univ. Pisa, 591 (1991). |
| [7] |
S. Luckhaus and T. Sturzenhecker,
Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), 253-271.
doi: 10.1007/BF01205007. |
| [8] |
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, volume 135 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2012. An
introduction to geometric measure theory.
doi: 10.1017/CBO9781139108133. |
| [9] |
M. Röger,
Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation, Interfaces Free Bound., 6 (2004), 105-133.
doi: 10.4171/IFB/93. |
| [10] |
R. Schätzle,
Hypersurfaces with mean curvature given by an ambient {S}obolev function, J. Differential Geom., 58 (2001), 371-420.
doi: 10.4310/jdg/1090348353. |
| [11] |
J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987),
65–96.
doi: 10.1007/BF01762360. |
| [12] |
L. Simon, Lectures on Geometric Measure Theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University., Australian National University Centre for Mathematical Analysis, Canberra, 1983. |
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