American Institute of Mathematical Sciences

January  2021, 14(1): 331-351. doi: 10.3934/dcdss.2020325

Existence of weak solutions for a sharp interface model for phase separation on biological membranes

 Faculty of Mathematics, University of Regensburg, 93040 Regensburg, Germany

* Corresponding author: Helmut Abels

Received  March 2019 Revised  September 2019 Published  January 2021 Early access  April 2020

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.

Citation: Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325
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. Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. E. Edwards, Functional Analysis, Dover Publications Inc. New York, 1995.  Google Scholar [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.  Google Scholar [6] S. Luckhaus, The Stefan problem with the Gibbs-Thomson law, Preprint Univ. Pisa, 591 (1991). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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. Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. E. Edwards, Functional Analysis, Dover Publications Inc. New York, 1995.  Google Scholar [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.  Google Scholar [6] S. Luckhaus, The Stefan problem with the Gibbs-Thomson law, Preprint Univ. Pisa, 591 (1991). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Feiyao Ma, Lihe Wang. Schauder type estimates of linearized Mullins-Sekerka problem. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1037-1050. doi: 10.3934/cpaa.2012.11.1037 [2] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (11) : 3973-3987. doi: 10.3934/dcdss.2020467 [3] Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381 [4] Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769 [5] Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032 [6] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 [7] Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431 [8] Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 [9] Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014 [10] Qigui Yang, Qiaomin Xiang. Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3267-3284. doi: 10.3934/dcdss.2020335 [11] Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087 [12] Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 [13] Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709 [14] Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005 [15] Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019 [16] Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045 [17] Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 [18] Hassan Belhadj, Mohamed Fihri, Samir Khallouq, Nabila Nagid. Optimal number of Schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 21-34. doi: 10.3934/dcdss.2018002 [19] Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021092 [20] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

2020 Impact Factor: 2.425