Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Special Issues: Nonlinear Evolution PDEs, Interfaces and Applications

We study the existence of weak solutions to a Cahn-Hilliard-Darcy system coupled witha convection-reaction-diffusion equation through the fluxes, through the source terms and in Darcy’slaw. The system of equations arises from a mixture model for tumour growth accounting for transportmechanisms such as chemotaxis and active transport. We prove, via a Galerkin approximation, theexistence of global weak solutions in two and three dimensions, along with new regularity results forthe velocity field and for the pressure. Due to the coupling with the Darcy system, the time derivativeshave lower regularity compared to systems without Darcy flow, but in the two dimensional case weemploy a new regularity result for the velocity to obtain better integrability and temporal regularityfor the time derivatives. Then, we deduce the global existence of weak solutions for two variantsof the model; one where the velocity is zero and another where the chemotaxis and active transportmechanisms are absent.
  Article Metrics

Keywords Cahn-Hilliard-Darcy system; phase field model; convection-reaction-diffusion equation; tumour growth; chemotaxis; weak solutions; asymptotic analysis

Citation: Harald Garcke, Kei Fong Lam. Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth. AIMS Mathematics, 2016, 1(3): 318-360. doi: 10.3934/Math.2016.3.318


  • 1. R.A. Adams and J.J.F. Fournier, Sobolev spaces. Pure and applied mathematics, Vol. 140. 2 Ed., Elsevier/Academic Press, Amsterdam, 2003.
  • 2. H.W. Alt, Linear Functional Analysis. An Application-Oriented Introduction. Translated from the German edition by Robert Nürnberg. Universitext. Springer Berlin London, 2016.
  • 3. S. Bosia, M. Conti, and M. Grasselli, On the Cahn-Hilliard-Brinkman system, Commun. Math. Sci., 13 (2015), no. 6, 1541-1567.
  • 4. E.A. Coddington and N. Levinson, Theory of Ordinary Di erential Equations. International series in pure and applied mathematics. Tata McGraw-Hill, New York, 1955.
  • 5. P. Colli, G. Gilardi, and D. Hilhorst, On a Cahn–Hilliard type phase field model related to tumor growth. Discrete Contin. Dyn. Syst., 35 (2015), no. 6, 2423-2442.
  • 6. P. Colli, G. Gilardi, E. Rocca, and J. Sprekels, Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth, Nonlinear Anal. Real World Appl., 26 (2015), 93-108.
  • 7. P. Colli, G. Gilardi, E. Rocca, and J. Sprekels, Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth, Discrete Contin. Dyn. Syst. Ser. S., in press (2016).
  • 8. V. Cristini and J. Lowengru, Multiscaled modeling of cancer. An Integrated Experiemental and Mathematical Modeling Approach. Cambridge University Press, 2010.
  • 9. M. Dai, E. Feireisl, E. Rocca, G. Schimperna, and M. Schonbek, Analysis of a di use interface model for multispecies tumor growth, Preprint arXiv:1507.07683 (2015).
  • 10. F. Della Porta and M. Grasselli, On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems, Commun. Pure Appl. Anal., 15 (2016), 299-317.
  • 11. E. DiBenedetto, Degenerate Parabolic Equations. Universitext. Springer–Verlag New York, 1993.
  • 12. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, Volume 19. AMS,Providence, Rhode Island, 2002.
  • 13. A. Fasano, A. Bertuzzi, and A. Gandolfi, Mathematical modeling of tumour growth and treatment.Complex Systems in Biomedicine. Springer Milan, 2006.
  • 14. X. Feng and S.Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shawflow and its fully discrete finite element approximation, SIAM J. Numer. Anal., 50 (2012), no. 3,1320-1343.
  • 15. A. Friedman, Partial Differential Equations. Holt, Rinehart and Winston, New York, 1969.
  • 16. S. Frigeri, M. Grasselli, and E. Rocca, On a diffuse interface model of tumor growth, European J.Appl. Math., 26 (2015), 215-243.
  • 17. H. Garcke and K.F. Lam, Well-posedness of a Cahn-Hilliard system modelling tumour growth withchemotaxis and active transport, European J. Appl. Math., in press (2016). DOI http://dx.doi.org/10.1017/S0956792516000292.
  • 18. H. Garcke and K.F. Lam, Analysis of a Cahn-Hilliard system with non zero Dirichlet conditionsmodelling tumour growth with chemotaxis, Preprint arXiv:1604.00287 (2016).
  • 19. H. Garcke, K.F. Lam, and E. Rocca, Optimal control of treatment time in a diffuse interface modelfor tumor growth, Preprint arXiv:1608.00488 (2016).
  • 20. H. Garcke, K.F. Lam, E. Sitka, and V. Styles, A Cahn-Hilliard-Darcy model for tumour growth withchemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), no. 6, 1095-1148.
  • 21. A. Hawkins-Daarud, K.G. van der Zee, and J.T. Oden, Numerical simulation of a thermodynamicallyconsistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012),3-24.
  • 22. J. Jiang, H.Wu, and S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth, J. Differential Equ., 259 (2015),no. 7, 3032-3077.
  • 23. H. Lee, J. Lowengrub, and J. Goodman, Modeling pinchoff and reconnection in a Hele–Shaw cell.I. The models and their calibration, Phys. Fluids, 14 (2002), no. 2, 492-513.
  • 24. H. Lee, J. Lowengrub, and J. Goodman, Modeling pinchoff and reconnection in a Hele-Shaw cell.II. Analysis and simulation in the nonlinear regime, Phys. Fluids, 14 (2002), no. 2, 514-545.
  • 25. J.S. Lowengrub, E. Titi, and K. Zhao, Analysis of a mixture model of tumor growth, European J.Appl. Math., 24 (2013), 691-734.
  • 26. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations. Texts in AppliedMathematics. 2 Eds., Springer-Verlag New York, 2004.
  • 27. H.L. Royden and P. Fitzpatrick, Real Analysis. Featured Titles for Real Analysis Series. 4 Eds.,Pearson Prentice Hall, Boston, 2010.
  • 28. J. Simon, Compact sets in space Lp(0; T; B), Ann. Mat. Pura Appl., 146 (1986), no. 1, 65-96.
  • 29. X. Wang and H. Wu, Long-time behavior for the Hele-Shaw-Cahn-Hilliard system, Asymptot.Anal., 78 (2012), no. 4, 217-245.
  • 30. X. Wang and Z. Zhang, Well-posedness of the Hele-Shaw-Cahn-Hilliard system, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 30 (2013), no. 3, 367-384.


This article has been cited by

  • 1. Harald Garcke, Kei Fong Lam, Elisabetta Rocca, Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth, Applied Mathematics & Optimization, 2017, 10.1007/s00245-017-9414-4
  • 2. Harald Garcke, Kei Fong Lam, Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete and Continuous Dynamical Systems, 2017, 37, 8, 4277, 10.3934/dcds.2017183
  • 3. Luca Dedè, Harald Garcke, Kei Fong Lam, A Hele–Shaw–Cahn–Hilliard Model for Incompressible Two-Phase Flows with Different Densities, Journal of Mathematical Fluid Mechanics, 2017, 10.1007/s00021-017-0334-5
  • 4. KEI FONG LAM, HAO WU, Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis, European Journal of Applied Mathematics, 2017, 1, 10.1017/S0956792517000298
  • 5. Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca, Sliding Mode Control for a Phase Field System Related to Tumor Growth, Applied Mathematics & Optimization, 2017, 10.1007/s00245-017-9451-z
  • 6. Sergio Frigeri, Kei Fong Lam, Elisabetta Rocca, , Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, 2017, Chapter 9, 217, 10.1007/978-3-319-64489-9_9
  • 7. Harald Garcke, Kei Fong Lam, Robert Nürnberg, Emanuel Sitka, A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis, Mathematical Models and Methods in Applied Sciences, 2018, 28, 03, 525, 10.1142/S0218202518500148
  • 8. Harald Garcke, Kei Fong Lam, , Trends in Applications of Mathematics to Mechanics, 2018, Chapter 12, 243, 10.1007/978-3-319-75940-1_12
  • 9. Andrea Giorgini, Maurizio Grasselli, Hao Wu, The Cahn–Hilliard–Hele–Shaw system with singular potential, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2018, 35, 4, 1079, 10.1016/j.anihpc.2017.10.002
  • 10. Andrea Signori, Optimal Distributed Control of an Extended Model of Tumor Growth with Logarithmic Potential, Applied Mathematics & Optimization, 2018, 10.1007/s00245-018-9538-1
  • 11. Matthias Ebenbeck, Harald Garcke, Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis, Journal of Differential Equations, 2018, 10.1016/j.jde.2018.10.045
  • 12. Jürgen Sprekels, Hao Wu, Optimal Distributed Control of a Cahn–Hilliard–Darcy System with Mass Sources, Applied Mathematics & Optimization, 2019, 10.1007/s00245-019-09555-4
  • 13. Cecilia Cavaterra, Elisabetta Rocca, Hao Wu, Long-Time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth, Applied Mathematics & Optimization, 2019, 10.1007/s00245-019-09562-5
  • 14. Alain Miranville, Elisabetta Rocca, Giulio Schimperna, On the long time behavior of a tumor growth model, Journal of Differential Equations, 2019, 10.1016/j.jde.2019.03.028
  • 15. Luca Dedè, Alfio Quarteroni, Isogeometric Analysis of a Phase Field Model for Darcy Flows with Discontinuous Data, Chinese Annals of Mathematics, Series B, 2018, 39, 3, 487, 10.1007/s11401-018-0079-3
  • 16. Matthias Ebenbeck, Harald Garcke, On a Cahn--Hilliard--Brinkman Model for Tumor Growth and Its Singular Limits, SIAM Journal on Mathematical Analysis, 2019, 51, 3, 1868, 10.1137/18M1228104
  • 17. Marvin Fritz, Ernesto A. B. F. Lima, J. Tinsley Oden, Barbara Wohlmuth, On the Unsteady Darcy-Forchheimer-Brinkman Equation in Local and Nonlocal Tumor Growth Models, Mathematical Models and Methods in Applied Sciences, 2019, 10.1142/S0218202519500325
  • 18. Matthias Ebenbeck, Patrik Knopf, Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation, Calculus of Variations and Partial Differential Equations, 2019, 58, 4, 10.1007/s00526-019-1579-z
  • 19. Marvin Fritz, Ernesto A. B. F. Lima, Vanja Nikolic, J. Tinsley Oden, Barbara Wohlmuth, Local and Nonlocal Phase-Field Models of Tumor Growth and Invasion Due to ECM Degradation, Mathematical Models and Methods in Applied Sciences, 2019, 10.1142/S0218202519500519
  • 20. Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels, A Distributed Control Problem for a Fractional Tumor Growth Model, Mathematics, 2019, 7, 9, 792, 10.3390/math7090792
  • 21. Andrea Signori, Vanishing parameter for an optimal control problem modeling tumor growth, Asymptotic Analysis, 2019, 1, 10.3233/ASY-191546
  • 22. Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels, Asymptotic analysis of a tumor growth model with fractional operators, Asymptotic Analysis, 2019, 1, 10.3233/ASY-191578
  • 23. Pierluigi Colli, Andrea Signori, Jürgen Sprekels, Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials, Applied Mathematics & Optimization, 2019, 10.1007/s00245-019-09618-6
  • 24. Michele Colturato, Sliding mode control for a diffuse interface tumor growth model coupling a Cahn-Hilliard equation with a reaction-diffusion equation, Mathematical Methods in the Applied Sciences, 2020, 10.1002/mma.6403
  • 25. Matthias Ebenbeck, Kei Fong Lam, Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms, Advances in Nonlinear Analysis, 2020, 10, 1, 24, 10.1515/anona-2020-0100
  • 26. Harald Garcke, Sema Yayla, Long-time dynamics for a Cahn–Hilliard tumor growth model with chemotaxis, Zeitschrift für angewandte Mathematik und Physik, 2020, 71, 4, 10.1007/s00033-020-01351-3
  • 27. Harald Garcke, Kei Fong Lam, Andrea Signori, On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects, Nonlinear Analysis: Real World Applications, 2021, 57, 103192, 10.1016/j.nonrwa.2020.103192

Reader Comments

your name: *   your email: *  

Copyright Info: 2016, Harald Garcke, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved