Open Access
Issue
ESAIM: M2AN
Volume 58, Number 5, September-October 2024
Page(s) 1989 - 2034
DOI https://doi.org/10.1051/m2an/2024063
Published online 21 October 2024
  1. H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194 (2009) 463–506. [Google Scholar]
  2. H. Abels, Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow. SIAM J. Math. Anal. 44 (2012) 316–340. [CrossRef] [MathSciNet] [Google Scholar]
  3. H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids. Indiana Univ. Math. J. 57 (2008) 659–698. [CrossRef] [Google Scholar]
  4. H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22 (2012) 40. [CrossRef] [Google Scholar]
  5. H. Abels, D. Depner and H. Garcke, On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility. Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013) 1175–1190. [CrossRef] [MathSciNet] [Google Scholar]
  6. A. Agosti, Discontinuous Galerkin finite element discretization of a degenerate Cahn–Hilliard equation with a single-well potential. Calcolo 56 (2019) 47. [CrossRef] [Google Scholar]
  7. A. Agosti, P.F. Antonietti, P. Ciarletta, M. Grasselli and M. Verani, A Cahn–Hilliard-type equation with application to tumor growth dynamics. Math. Methods Appl. Sci. 40 (2017) 7598–7626. [CrossRef] [MathSciNet] [Google Scholar]
  8. G.L. Aki, W. Dreyer, J. Giesselmann and C. Kraus, A quasi-incompressible diffuse interface model with phase transition. Math. Models Methods Appl. Sci. 24 (2014) 827–861. [CrossRef] [MathSciNet] [Google Scholar]
  9. D. Ambrosi and L. Preziosi, On the closure of mass balance models for tumor growth. Math. Models Methods Appl. Sci. 12 (2002) 737–754. [CrossRef] [MathSciNet] [Google Scholar]
  10. D.M. Anderson, G.B. McFadden and AA. Wheeler, Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1998) 139–165. [CrossRef] [Google Scholar]
  11. X. Antoine, J. Shen and Q. Tang, Scalar auxiliary variable/Lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations. J. Comput. Phys. 437 (2021) 19. [Google Scholar]
  12. J.W. Barrett, J.F. Blowey and H. Garcke, Finite element approximation of the Cahn–Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286–318. [CrossRef] [MathSciNet] [Google Scholar]
  13. D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238 (2003) 211–223. [Google Scholar]
  14. D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier–Stokes models. J. Math. Pures Appl. 86 (2006) 362–368. [CrossRef] [MathSciNet] [Google Scholar]
  15. D. Bresch, A.F. Vasseur and C. Yu, Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities. J. Eur. Math. Soc. (JEMS) 24 (2022) 1791–1837. [Google Scholar]
  16. H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20 (2003) 341–366. [CrossRef] [PubMed] [Google Scholar]
  17. J.W. Cahn, On spinodal decomposition. Acta metall. 9 (1961) 795–801. [CrossRef] [Google Scholar]
  18. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. [CrossRef] [Google Scholar]
  19. C. Chatelain, T. Balois, P. Ciarletta and M.B. Amar, Emergence of microstructural patterns in skin cancer: a phase separation analysis in a binary mixture. New J. Phys. 13 (2011) 115013. [CrossRef] [Google Scholar]
  20. C. Chatelain, P. Ciarletta and M. Ben Amar, Morphological changes in early melanoma development: influence of nutrients, growth inhibitors and cell-adhesion mechanisms. J. Theoret. Biol. 290 (2011) 46–59. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  21. L. Cherfils, E. Feireisl, M. Michálek, A. Miranville, M. Petcu and D. Pražák, The compressible Navier–Stokes–Cahn–Hilliard equations with dynamic boundary conditions. Math. Models Methods Appl. Sci. 29 (2019) 2557–2584. [CrossRef] [MathSciNet] [Google Scholar]
  22. M. Ebenbeck, H. Garcke and R. Nürnberg, Cahn–Hilliard–Brinkman systems for tumour growth. Discrete Cont. Dyn. S 14 (2021) 3989–4033. [CrossRef] [Google Scholar]
  23. C. Eck, H. Garcke and P. Knabner, Mathematical Modeling. Undergraduate Mathematics Series. Springer, Cham (2017). [CrossRef] [Google Scholar]
  24. 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]
  25. J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113 (1990) 97–120. [Google Scholar]
  26. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis. Vol. VII. North-Holland, Amsterdam (2000) 713–1020. [Google Scholar]
  27. E. Feireisl, Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and its Applications. Vol. 26. Oxford University Press, Oxford (2004). [Google Scholar]
  28. E. Feireisl and P. Laurençot, Non-isothermal Smoluchowski-Poisson equations as a singular limit of the Navier–Stokes-Fourier-Poisson system. J. Math. Pures Appl. 88 (2007) 325–349. [CrossRef] [MathSciNet] [Google Scholar]
  29. E. Feireisl and A. Novotný, Singular limits in thermodynamics of viscous fluids, in Advances in Mathematical Fluid Mechanics. Birkh¨auser Verlag, Basel (2009). [Google Scholar]
  30. M. Fritz, Tumor evolution models of phase-field type with nonlocal effects and angiogenesis. Bull. Math. Biol. 85 (2023) 34. [CrossRef] [PubMed] [Google Scholar]
  31. H. Garcke, K.F. Lam, E. Sitka and V. Styles, A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport. Math. Models Methods Appl. Sci. 26 (2016) 1095–1148. [CrossRef] [MathSciNet] [Google Scholar]
  32. J. Giesselmann and T. Pryer, Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model. ESAIM Math. Model. Numer. Anal. 49 (2015) 275–301. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  33. Z. Guo, P. Lin and J.S. Lowengrub, A numerical method for the quasi-incompressible Cahn–Hilliard–Navier–Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276 (2014) 486–507. [CrossRef] [MathSciNet] [Google Scholar]
  34. M.E. Gurtin, On a nonequilibrium thermodynamics of capillarity and phase. Q. Appl. Math. 47 (1989) 129–145. [CrossRef] [Google Scholar]
  35. M.E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6 (1996) 815–831. [CrossRef] [Google Scholar]
  36. M.E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010). [CrossRef] [Google Scholar]
  37. Q. He and X. Shi, Numerical study of compressible Navier–Stokes-Cahn–Hilliard system. Commun. Math. Sci. 18 (2020) 571–591. [CrossRef] [MathSciNet] [Google Scholar]
  38. P.C. Hohenberg and B.I. Halperin, Theory of dynamic critical phenomena. Rev. Mod. Phys. 49 (1977) 435–479. [CrossRef] [Google Scholar]
  39. B.S. Hosseini, S. Turek, M. Möller and C. Palmes, Isogeometric analysis of the Navier–Stokes–Cahn–Hilliard equations with application to incompressible two-phase flows. J. Comput. Phys. 348 (2017) 171–194. [CrossRef] [MathSciNet] [Google Scholar]
  40. F. Huang and J. Shen, Bound/positivity preserving and energy stable scalar auxiliary variable schemes for dissipative systems: applications to Keller–Segel and Poisson–Nernst–Planck equations. SIAM J. Sci. Comput. 43 (2021) A1832–A1857. [CrossRef] [Google Scholar]
  41. F. Huang and J. Shen, A new class of implicit-explicit BDFk SAV schemes for general dissipative systems and their error analysis. Comput. Methods Appl. Mech. Eng. 392 (2022) 25. [Google Scholar]
  42. F. Huang, J. Shen and K. Wu, Bound/positivity preserving and unconditionally stable schemes for a class of fourth order nonlinear equations. J. Comput. Phys. 460 (2022) 16. [Google Scholar]
  43. M. Jiang, Z. Zhang and J. Zhao, Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation. J. Comput. Phys. 456 (2022) 20. [Google Scholar]
  44. S. Jin and Z.P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48 (1995) 235–276. [CrossRef] [Google Scholar]
  45. K.F. Lam and H. Wu, Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis. Eur. J. Appl. Math. 29 (2018) 595–644. [CrossRef] [Google Scholar]
  46. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Vol. 31. Cambridge University Press, Cambridge (2002). [CrossRef] [Google Scholar]
  47. X. Li and J. Shen, On a SAV-MAC scheme for the Cahn–Hilliard–Navier–Stokes phase-field model and its error analysis for the corresponding Cahn–Hilliard–Stokes case. Math. Models Methods Appl. Sci. 30 (2020) 2263–2297. [CrossRef] [MathSciNet] [Google Scholar]
  48. P.-L. Lions, Mathematical Topics in Fluid Mechanics: Vol. 2: Compressible Models. Vol. 10 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1998). [Google Scholar]
  49. S. Lisini, D. Matthes and G. Savaré, Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics. J. Differ. Equ. 253 (2012) 814–850. [CrossRef] [Google Scholar]
  50. I.S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rational Mech. Anal. 46 (1972) 131–148. [CrossRef] [MathSciNet] [Google Scholar]
  51. J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn–Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998) 2617–2654. [CrossRef] [MathSciNet] [Google Scholar]
  52. J. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth. Eur. J. Appl. Math. 24 (2013) 691–734. [CrossRef] [Google Scholar]
  53. A. Miranville, The Cahn–Hilliard Equation: Recent Advances and Applications. Vol. 995 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2019). [Google Scholar]
  54. A. Moussa, Some variants of the classical Aubin-Lions lemma. J. Evol. Equ. 16 (2016) 65–93. [CrossRef] [MathSciNet] [Google Scholar]
  55. I. Müller, Thermodynamics. Pitman, Boston (1985). [Google Scholar]
  56. G.A. Narsilio, O. Buzzi, S. Fityus, T.S. Yun and D.W. Smith, Upscaling of Navier–Stokes equations in porous media: theoretical, numerical and experimental approach. Comput. Geotechn. 36 (2009) 1200–1206. [CrossRef] [Google Scholar]
  57. B. Perthame and A. Poulain, Relaxation of the Cahn–Hilliard equation with singular single-well potential and degenerate mobility. Eur. J. Appl. Math. 32 (2021) 89–112. [CrossRef] [Google Scholar]
  58. A. Poulain and K. Schratz, Convergence, error analysis and longtime behavior of the scalar auxiliary variable method for the nonlinear Schrödinger equation. IMA J. Numer. Anal. 42 (2022) 2853–2883. [CrossRef] [MathSciNet] [Google Scholar]
  59. 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. [CrossRef] [Google Scholar]
  60. J. Shen, J. Xu and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353 (2018) 407–416. [CrossRef] [MathSciNet] [Google Scholar]
  61. 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]
  62. J. Shen and X. Yang, The IEQ and SAV approaches and their extensions for a class of highly nonlinear gradient flow systems. Contemp. Math. 754 (2020) 217–245. [CrossRef] [Google Scholar]
  63. M.F.P. ten Eikelder, K.G. van der Zee, I. Akkerman and D. Schillinger, A unified framework for Navier–Stokes Cahn–Hilliard models with non-matching densities. Math. Models Methods Appl. Sci. 33 (2023) 175–221. [CrossRef] [MathSciNet] [Google Scholar]
  64. A.F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations. Invent. Math. 206 (2016) 935–974. [CrossRef] [MathSciNet] [Google Scholar]
  65. S.M. Wise, J.S. Lowengrub, H.B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth – I: model and numerical method. J. Theor. Biol. 253 (2008) 524–543. [CrossRef] [Google Scholar]
  66. Y. Zhang and J. Shen, A generalized SAV approach with relaxation for dissipative systems. J. Comput. Phys. 464 (2022) 23. [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.

Recommended for you