Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation

Martin Heida

Applications of Mathematics (2015)

  • Volume: 60, Issue: 1, page 51-90
  • ISSN: 0862-7940

Abstract

top
We show existence of solutions to two types of generalized anisotropic Cahn-Hilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical no-flux boundary conditions where the mobility depends on concentration u , gradient of concentration ∇ u and the chemical potential Δ u - s ' ( u ) . The existence is shown using a newly developed generalization of gradient flows by the author and the theory of Young measures.

How to cite

top

Heida, Martin. "Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation." Applications of Mathematics 60.1 (2015): 51-90. <http://eudml.org/doc/262195>.

@article{Heida2015,
abstract = {We show existence of solutions to two types of generalized anisotropic Cahn-Hilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical no-flux boundary conditions where the mobility depends on concentration $u$, gradient of concentration $\nabla u$ and the chemical potential $\Delta u-s^\{\prime \}(u)$. The existence is shown using a newly developed generalization of gradient flows by the author and the theory of Young measures.},
author = {Heida, Martin},
journal = {Applications of Mathematics},
keywords = {Cahn-Hilliard; anisotropic behavior; gradient flow; curve of maximal slope; entropy; Cahn-Hilliard equation; anisotropic behavior; gradient flow; curve of maximal slope; entropy},
language = {eng},
number = {1},
pages = {51-90},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation},
url = {http://eudml.org/doc/262195},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Heida, Martin
TI - Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 51
EP - 90
AB - We show existence of solutions to two types of generalized anisotropic Cahn-Hilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical no-flux boundary conditions where the mobility depends on concentration $u$, gradient of concentration $\nabla u$ and the chemical potential $\Delta u-s^{\prime }(u)$. The existence is shown using a newly developed generalization of gradient flows by the author and the theory of Young measures.
LA - eng
KW - Cahn-Hilliard; anisotropic behavior; gradient flow; curve of maximal slope; entropy; Cahn-Hilliard equation; anisotropic behavior; gradient flow; curve of maximal slope; entropy
UR - http://eudml.org/doc/262195
ER -

References

top
  1. Abels, H., Wilke, M., 10.1016/j.na.2006.10.002, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67 3176-3193 (2007). (2007) Zbl1121.35018MR2347608DOI10.1016/j.na.2006.10.002
  2. Adams, R. A., Sobolev Spaces, Pure and Applied Mathematics 65. A Series of Monographs and Textbooks Academic Press, New York (1975). (1975) Zbl0314.46030MR0450957
  3. Brokate, M., Kenmochi, N., Müller, I., Rodriguez, J. F., Verdi, C., Phase transitions and hysteresis, Lectures Given at the Third C.I.M.E., 1993, Montecatini Terme, Italy Lecture Notes in Mathematics 1584 Springer, Berlin (1994), A. Visintin. (1994) MR1321829
  4. Buscaglia, G. C., Ausas, R. F., 10.1016/j.cma.2011.06.002, Comput. Methods Appl. Mech. Eng. 200 3011-3025 (2011). (2011) Zbl1230.76047MR2844033DOI10.1016/j.cma.2011.06.002
  5. Cahn, J. W., Elliott, C. M., Novick-Cohen, A., 10.1017/S0956792500002369, Eur. J. Appl. Math. 7 287-301 (1996). (1996) Zbl0861.35039MR1401172DOI10.1017/S0956792500002369
  6. Changchun, L., Cahn-Hilliard equation with terms of lower order and non-constant mobility, Electron. J. Qual. Theory Differ. Equ. 2003 (2003), 9 pp. (electronic only). (2003) Zbl1032.35076MR1986908
  7. Passo, R. Dal, Giacomelli, L., Novick-Cohen, A., 10.4171/IFB/9, Interfaces Free Bound. 1 199-226 (1999). (1999) MR1867131DOI10.4171/IFB/9
  8. Debussche, A., Dettori, L., 10.1016/0362-546X(94)00205-V, Nonlinear Anal., Theory Methods Appl. 24 1491-1514 (1995). (1995) Zbl0831.35088MR1327930DOI10.1016/0362-546X(94)00205-V
  9. Dellacherie, C., Meyer, P.-A., Probabilities and Potential, North-Holland Mathematics Studies 29 North-Holland Publishing Company, Amsterdam (1978). (1978) Zbl0494.60001MR0521810
  10. Elliott, C. M., Garcke, H., 10.1137/S0036141094267662, SIAM J. Math. Anal. 27 404-423 (1996). (1996) Zbl0856.35071MR1377481DOI10.1137/S0036141094267662
  11. Elliott, C. M., Zheng, S., 10.1007/BF00251803, Arch. Ration. Mech. Anal. 96 339-357 (1986). (1986) Zbl0624.35048MR0855754DOI10.1007/BF00251803
  12. Gal, C. G., Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differ. Equ. 12 1241-1274 (2007). (2007) Zbl1162.35386MR2372239
  13. Gilardi, G., Miranville, A., Schimperna, G., 10.3934/cpaa.2009.8.881, Commun. Pure Appl. Anal. 8 881-912 (2009). (2009) Zbl1172.35417MR2476663DOI10.3934/cpaa.2009.8.881
  14. Grasselli, M., Miranville, A., Rossi, R., Schimperna, G., 10.1080/03605302.2010.543945, Commun. Partial Differ. Equations 36 1193-1238 (2011). (2011) Zbl1241.35024MR2810586DOI10.1080/03605302.2010.543945
  15. Heida, M., Modeling Multiphase Flow in Porous Media with an Application to Permafrost Soil, Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät, Heidelberg; Charles Univ. Praha, Faculty of Mathematics and Physics (PhD Thesis), Praha (2011). (2011) 
  16. Heida, M., 10.1016/j.ijengsci.2012.09.005, Internat. J. Engrg. Sci. 62 126-156 (2013). (2013) MR2996309DOI10.1016/j.ijengsci.2012.09.005
  17. Heida, M., 10.1016/j.jmaa.2014.09.046, J. Math. Anal. Appl. 423 410-455 (2015). (2015) MR3273188DOI10.1016/j.jmaa.2014.09.046
  18. Ito, A., Kenmochi, N., Niezgódka, M., Large-time behaviour of non-isothermal models for phase separation, Proc. Conf. Elliptic and Parabolic Problems, 1994 Pitman Res. Notes Math. Ser. 325 Longman Scientific & Technical, Harlow 120-151 (1995). (1995) Zbl0838.35053MR1416579
  19. Lisini, S., Matthes, D., Savaré, G., 10.1016/j.jde.2012.04.004, J. Differ. Equations 253 814-850 (2012). (2012) Zbl1248.35095MR2921215DOI10.1016/j.jde.2012.04.004
  20. Liu, C., 10.1016/j.jmaa.2008.02.027, J. Math. Anal. Appl. 344 124-144 (2008). (2008) Zbl1158.35077MR2416296DOI10.1016/j.jmaa.2008.02.027
  21. Liu, C., Qi, Y., Yin, J., 10.1007/s10114-005-0711-5, Acta Math. Sin., Engl. Ser. 22 1139-1150 (2006). (2006) Zbl1106.35011MR2245245DOI10.1007/s10114-005-0711-5
  22. Lowengrub, J. S., Rätz, A., Voigt, A., 10.1103/PhysRevE.79.031926, Phys. Rev. E 79 (2009), 031926, 13 pp. (2009) MR2497179DOI10.1103/PhysRevE.79.031926
  23. Mercker, M., Models, numerics and simulations of deforming biological surfaces, Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät (PhD Thesis), Heidelberg (2012). (2012) Zbl1304.92023
  24. Mercker, M., Marciniak-Czochra, A., Richter, T., Hartmann, D., 10.1137/120885553, SIAM J. Appl. Math. 73 1768-1792 (2013). (2013) Zbl1282.74015MR3095724DOI10.1137/120885553
  25. Mercker, M., Richter, T., Hartmann, D., 10.1021/jp204127g, J. Phys. Chem. B 115 11739-11745, DOI: 10.1021/jp204127g (2011). (2011) DOI10.1021/jp204127g
  26. Miranville, A., Existence of solutions for Cahn-Hilliard type equations, Discrete Contin. Dyn. Syst. 2003 Suppl. Vol., 630-637 (2003). (2003) Zbl1070.35002MR2018168
  27. Miranville, A., Zelik, S., 10.1002/mma.464, Math. Methods Appl. Sci. 27 545-582 (2004). (2004) Zbl1050.35113MR2041814DOI10.1002/mma.464
  28. Miranville, A., Zelik, S., 10.3934/dcds.2010.28.275, Discrete Contin. Dyn. Syst. 28 275-310 (2010). (2010) Zbl1203.35046MR2629483DOI10.3934/dcds.2010.28.275
  29. Mugnai, L., Röger, M., 10.4171/IFB/179, Interfaces Free Bound. 10 45-78 (2008). (2008) Zbl1288.93096MR2383536DOI10.4171/IFB/179
  30. Mugnai, L., Röger, M., 10.1512/iumj.2011.60.3949, Indiana Univ. Math. J. 60 41-76 (2011). (2011) MR2952409DOI10.1512/iumj.2011.60.3949
  31. Novick-Cohen, A., The Cahn-Hilliard equation, Handbook of Differential Equations: Evolutionary Equations. Vol. IV Elsevier/North-Holland, Amsterdam 201-228 (2008). (2008) Zbl1185.35001MR2508166
  32. Novick-Cohen, A., The Cahn-Hilliard Equation: From Backwards Diffusion to Surface Diffusion, (to appear) in Cambridge University Press. MR2508166
  33. Qian, T., Wang, X.-P., Sheng, P., 10.1017/S0022112006001935, J. Fluid Mech. 564 333-360 (2006). (2006) Zbl1178.76296MR2261865DOI10.1017/S0022112006001935
  34. Racke, R., Zheng, S., The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differ. Equ. 8 83-110 (2003). (2003) Zbl1035.35050MR1946559
  35. Röger, M., Schätzle, R., 10.1007/s00209-006-0002-6, Math. Z. 254 675-714 (2006). (2006) Zbl1126.49010MR2253464DOI10.1007/s00209-006-0002-6
  36. Roidos, N., Schrohe, E., 10.1080/03605302.2012.736913, Commun. Partial Differ. Equations 38 925-943 (2013). (2013) Zbl1272.58013MR3046298DOI10.1080/03605302.2012.736913
  37. Rossi, R., 10.3934/cpaa.2005.4.405, Commun. Pure Appl. Anal. 4 405-430 (2005). (2005) Zbl1078.35058MR2149524DOI10.3934/cpaa.2005.4.405
  38. Rossi, R., Savaré, G., 10.1051/cocv:2006013, ESAIM, Control Optim. Calc. Var. 12 564-614 (2006). (2006) Zbl1116.34048MR2224826DOI10.1051/cocv:2006013
  39. Serfaty, S., Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. 31 1427-1451 (2011). (2011) Zbl1239.35015MR2836361
  40. Stefanelli, U., 10.1137/070684574, SIAM J. Control Optim. 47 1615-1642 (2008). (2008) Zbl1194.35214MR2425653DOI10.1137/070684574
  41. Taylor, J. E., Cahn, J. W., 10.1007/BF02186838, J. Stat. Phys. 77 183-197 (1994). (1994) Zbl0844.35044MR1300532DOI10.1007/BF02186838
  42. Temam, R., Navier-Stokes Equations. Theory and Numerical Analysis, American Mathematical Society, Providence (2001). (2001) Zbl0981.35001MR1846644
  43. Torabi, S., Lowengrub, J., Voigt, A., Wise, S., 10.1098/rspa.2008.0385, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465 1337-1359 (2009). (2009) Zbl1186.80014MR2500806DOI10.1098/rspa.2008.0385
  44. Torabi, S., Wise, S., Lowengrub, J., Rätz, A., Voigt, A., A new method for simulating strongly anisotropic Cahn-Hilliard equations, Materials Science and Technology-Association for Iron and Steel Technology 3 1432-1444 (2007). (2007) 
  45. Wise, S., Kim, J., Lowengrub, J., 10.1016/j.jcp.2007.04.020, J. Comput. Phys. 226 414-446 (2007). (2007) Zbl1310.82044MR2356365DOI10.1016/j.jcp.2007.04.020

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.