AIMS Mathematics, 2016, 1(3): 318-360. doi: 10.3934/Math.2016.3.318 Research article Special Issues

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

Harald Garcke, Kei Fong Lam

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Received: , Accepted: , Published:

Special Issues: Nonlinear Evolution PDEs, Interfaces and Applications

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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.

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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 L^p(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.

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)

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