Open Access
Review
Issue |
ESAIM: M2AN
Volume 59, Number 3, May-June 2025
|
|
---|---|---|
Page(s) | 1685 - 1704 | |
DOI | https://doi.org/10.1051/m2an/2025039 | |
Published online | 26 June 2025 |
- J.W. Barrett, J.F. Blowey and H. Garcke, Finite approximation of the Cahn–Hilliard equation with degenerate mobility. SIAM J Numer. Anal. 37 (2000) 286–318. [Google Scholar]
- J.W. Barrett, H. Garcke and R. Nürnberg, Finite element approximation for the dynamics of fluidic two-phase biomembranes. ESAIM: M2AN 51 (2017) 2319–2366. [CrossRef] [EDP Sciences] [Google Scholar]
- A. Bianchi, K. Painter and J. Sherratt, A mathematical model for lymphangiogenesis in normal and diabetic wounds. J. Theor. Biol. 383 (2015) 61–86. [Google Scholar]
- A. Bianchi, K. Painter and J. Sherratt, Spatio-temporal models of lymphangiogenesis in wound healing. Bull. Math. Biol. 78 (2016) 1904–1941. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- K. Boardman and M. Swartz, Interstitial flow as a guide for lymphatics. Circ. Res. 92 (2003) 801–808. [CrossRef] [PubMed] [Google Scholar]
- J. Cauvin-Vila, V. Ehrlacher, G. Marino and J.-F. Pietschmann, Stationary solutions and large time asymptotics to a cross-diffusion-Cahn–Hilliard system. Nonlinear Anal. 241 (2024) 113482. [Google Scholar]
- W. Chen, C. Wang, X. Wang and S. Wise, Positivity preserving, energy stable numerical schemes for the Cahn–Hilliard equation with logarithmic potential. J. Comput. Phys. X 3 (2019) 100031. [Google Scholar]
- S. Dai and Q. Du, Weak solutions for the Cahn–Hilliard equation with degenerate mobility. Arch. Ration. Mech. Anal. 219 (2016) 1161–1184. [CrossRef] [MathSciNet] [Google Scholar]
- A. Diegel, C. Wang and S. Wise, Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation. IMA J. Numer. Anal. 36 (2016) 1867–1897. [Google Scholar]
- V. Ehrlacher, G. Marino and J.-F. Pietschmann, Existence of weak solutions to a cross-diffusion Cahn–Hilliard type system. J. Differ. Equ. 286 (2021) 578–623. [Google Scholar]
- C.M. Elliott and H. Garcke, On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (1996) 404–423. [CrossRef] [MathSciNet] [Google Scholar]
- C.M. Elliott and A. Stuart, The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30 (1993) 1622–1663. [Google Scholar]
- D. Eyre, Unconditionally gradient stable time marching the Cahn–Hilliard equation. MRS Online Proc. Library 529 (1998) 39–47. [Google Scholar]
- P. Flory, Thermodynamics of high polymer solutions. J. Chem. Phys. 10 (1942) 51–61. [Google Scholar]
- A. Friedman and G. Lolas, Analysis of a mathematical model of tumor lymphangiogenesis. Math. Models Meth. Appl. Sci. 15 (2005) 95–107. [Google Scholar]
- Z. Fu and J. Yang, Energy-decreasing exponential time differencing Runge–Kutta methods for phase-field models. J. Comput. Phys. 454 (2022) 110943. [Google Scholar]
- Z. Fu, T. Tang and J. Yang, Energy preserving implicit-explicit Runge–Kutta methods for gradient flows. Math. Comput. 93 (2024) 2745–2767. [Google Scholar]
- D. Furihata, A stable and conservative finite difference scheme for the Cahn–Hilliard equation. Numer. Math. 87 (2001) 675–699. [Google Scholar]
- M. Gao and X.-P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem. J. Comput. Phys. 231 (2012) 1372–1386. [Google Scholar]
- H. Garcke, J. Kampmann, A. R¨atz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes. Math. Models Meth. Appl. Sci. 26 (2016) 1149–1189. [Google Scholar]
- H. Garcke, B. Kovács and D. Trautwein, Viscoelastic Cahn–Hilliard models for tumour growth. Math. Meth. Appl. Sci. 32 (2022) 2673–2758. [Google Scholar]
- F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. [CrossRef] [MathSciNet] [Google Scholar]
- M. Huggins, Solutions of long-chain compounds. J. Chem. Phys. 9 (1941) 440. [Google Scholar]
- X. Huo, A. Jüngel and A. Tzavaras, Existence and weak-strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems. Ann. Inst. H. Poincaré Anal. Non Lin. 41 (2023) 797–852. [CrossRef] [Google Scholar]
- A. Jüngel and Y. Li, Global weak solutions to a fractional Cahn–Hilliard cross-diffusion system in lymphangiogenesis. Preprint arXiv:2408.05972 (2024). [Google Scholar]
- A. Jüngel and Y. Li, Existence of global weak solutions to a Cahn–Hilliard cross-diffusion system in lymphangiogenesis. Discrete Contin. Dyn. Syst. 45 (2025) 286–308. [CrossRef] [MathSciNet] [Google Scholar]
- A. Jüngel and B. Wang, Structure-preserving semi-convex-splitting numerical scheme for a Cahn–Hilliard cross-diffusion system in lymphangiogenesis. Math. Meth. Appl. Sci. 34 (2023) 1905–1932. [Google Scholar]
- S. Lee and J. Shin, Energy stable compact scheme for Cahn–Hilliard equation with periodic boundary condition. Comput. Math. Appl. 77 (2019) 189–198. [CrossRef] [MathSciNet] [Google Scholar]
- H.-L. Liao, B. Ji, L. Wang and Z. Zhang, Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn–Hilliard model. J. Sci. Comput. 92 (2022) 52. [Google Scholar]
- Q.-X. Liu, A. Doelman, V. Rottsch¨afer, M. de Jager, P.M.J. Herman, M. Rietkerk and J. van de Koppel, Phase separation explains a new class of self-organized spatial patterns in ecological systems. Proc. Nat. Acad. Sci. 110 (2013) 11905–11910. [Google Scholar]
- K. Margaris and R. Black, Modelling the lymphatic system: challenges and opportunities. J. R. Soc. Interface 9 (2012) 601–612. [Google Scholar]
- D. Matthes and J. Zinsl, Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type. Nonlin. Anal. 159 (2017) 316–338. [Google Scholar]
- R.L. Pego, Front migration in the nonlinear Cahn–Hilliard equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 422 (1989) 261–278. [Google Scholar]
- E. Rocca, G. Schimperna and A. Signori, On a Cahn–Hilliard–Keller–Segel model with generalized logistic source describing tumor growth. J. Differ. Equ. 343 (2023) 530–578. [Google Scholar]
- T. Roose and A. Fowler, Network development in biological gels: role in lymphatic vessel development. Bull. Math. Biol. 70 (2008) 1772–1789. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- T. Roose and M. Swartz, Multiscale modeling of lymphatic drainage from tissues using homogenization theory. J. Biomech. 45 (2011) 107–115. [Google Scholar]
- J. Rutkowski, K. Boardman and M. Swartz, Characterization of lymphangiogenesis in a model of adult skin regeneration. Amer. J. Physiol. Heart Circ. Physiol. 291 (2006) H1402–H1410. [CrossRef] [PubMed] [Google Scholar]
- M. Scianna, C. Bell and L. Preziosi, A review of mathematical models for the formation of vascular networks. J. Theor. Biol. 333 (2013) 174–209. [Google Scholar]
- J. Shen, J. Xu and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353 (2018) 407–416. [Google Scholar]
- J. Shen, J. Xu and J. Yang, A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61 (2019) 474–506. [CrossRef] [MathSciNet] [Google Scholar]
- T. Tammela and K. Alitalo, Lymphangiogenesis: molecular mechanisms and future promise. Cell 140 (2010) 460–476. [CrossRef] [PubMed] [Google Scholar]
- T. Tang, X. Wu and J. Yang, Arbitrarily high order and fully discrete extrapolated RK–SAV/DG schemes for phase-field gradient flows. J. Sci. Comput. 93 (2022) 38. [Google Scholar]
- K. Wertheim and T. Roose, A mathematical model of lymphangiogenesis in a zebrafish embryo. Bull. Math. Biol. 79 (2017) 693–737. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.