In this paper, we consider a class of the degenerate Cahn–Hilliard equation with a smooth double-well potential. By applying a semi-discretization technique and asymptotic analysis method on non-degenerate Cahn–Hilliard equation, we obtain the existence and regularity of weak solutions to the Cahn–Hilliard equation with degenerate mobility. Moreover, we define a new entropy and obtain the global entropy estimates.
H.Abels and M.Wilke, Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy, Nonlinear Anal.67 (2007), 3176–3193. doi:10.1016/j.na.2006.10.002.
2.
A.L.Bertozzi and M.Puch, The lubrication approximation for thin viscous films: Regularity and Long-time behavior of weak solutions, Comm. Pure Appl. Math. (1996), 85–132. doi:10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2.
3.
J.F.Blowey and C.M.Elliott, The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part I: Mathematical analysis, Euro. Jnl of Applied Mathematics.2 (1991), 233–279. doi:10.1017/S095679250000053X.
4.
D.Brochet, D.Hilhorst and A.Novick-Cohen, Maximal attrator and inertial sets for a conserved phase field model, Adv. Diff. Eqns. (1991), 547–578.
5.
J.W.Cahn and J.Hilliard, Free energy of a nonuniform system I, interficial free energy, J. Chem. Phs.28 (1958), 258–267. doi:10.1063/1.1744102.
6.
J.W.Cahn, On spinodal decomposition, Acta Metall.9 (1961), 795–801. doi:10.1016/0001-6160(61)90182-1.
7.
G.Chamoun, M.Saad and R.Talhouk, A coupled anisotropic chemotaxis-fluid model: The case of two-sidedly degenerate diffusion, Computers and Mathematics with Applications. (2014), 1052–1070. doi:10.1016/j.camwa.2014.04.010.
8.
D.Cohen and J.M.Murray, A generalized diffusion model for growth and dispersion in a population, J. Math. Biol.12 (1981), 237–248. doi:10.1007/BF00276132.
9.
S.B.Dai and Q.Du, Weak solutions for the Cahn–Hilliard equation with degenerate mobility, Arch. Rational Mech. Anal.219 (2016), 1161–1184. doi:10.1007/s00205-015-0918-2.
10.
S.B.Dai and Q.Du, Coarening mechanism for systems governed by the Cahn–Hilliard equation with degenerate diffusion mobility, Multiscale Model. Simul.12 (2014), 1870–1890. doi:10.1137/140952387.
11.
C.M.Elliott and H.Garcke, On the Cahn–Hilliard equation with degernerate mobility, Siam J. Math. Anal.27 (1996), 404–423. doi:10.1137/S0036141094267662.
D.Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Spring-Verlag, New York, 1981, p. 840.
14.
E.Khain and L.M.Sander, A generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E.77 (2008), 051129. doi:10.1103/PhysRevE.77.051129.
15.
I.Klapper and J.Dockery, Role of cohesion in the material description of biofilms, Phys. Rev. E.74 (2006), 21.
16.
D.Li and C.Zhong, Global attractor for the Cahn–Hilliard system with fast growing nonlinearity, J. Diff. Eqns.149 (1998), 191–210. doi:10.1006/jdeq.1998.3429.
17.
S.Lisini, D.Matthes and G.Savaré, Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wassertein, J. Diff. Eqns.253 (2012), 814–850. doi:10.1016/j.jde.2012.04.004.
18.
F.Marpeau and M.Saad, Mathematical analysis of radionuclides displacement in porous media with nonlinear adsorption, J. Diff. Eqns.228 (2006), 412–439. doi:10.1016/j.jde.2006.03.023.
A.Miranville, Asymptotic behaviour of a generalized Cahn–Hilliard equation with a proliferation term, Applicable Analysis.92 (2013), 1308–1321. doi:10.1080/00036811.2012.671301.
21.
A.Miranville, The Cahn–Hilliard equation and some of its variants,AIMS, Mathematics.2 (2017), 479–544.
22.
A.Miranville and S.Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in: Handbook of Differential Equations, Evolutionary Partial Differential Equations, C.M.Dafermos and M.Pokorny, eds, Vol. 4, Elsevier, Amsterdam, 2008, pp. 103–200.
23.
A.Miranville and S.Zelik, The Cahn–Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Systems.28 (2010), 275–310. doi:10.3934/dcds.2010.28.275.
24.
A.Novick-Cohen and L.A.Peletier, The steady states of the one-dimensional Cahn–Hilliard equation, Appl. Math. Lett.3 (1992), 45–46. doi:10.1016/0893-9659(92)90036-9.
25.
A.Novick-Cohen, On Cahn–Hilliard type equations, Nonlinear Anal TMA.15 (1990), 797–814. doi:10.1016/0362-546X(90)90094-W.
26.
A.Novick-Cohen, The Cahn–Hilliard Equation, Handbook of Differential Equations- Evolutionary Equation Volume 4. North-Holland Is an Imprint of, Elsevier, UK, 2008.
27.
R.D.Passo, H.Garcke and G.Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, Siam J. Math. Anal.29 (1998), 321–342. doi:10.1137/S0036141096306170.
28.
G.Schimperna, Global attractor for Cahn–Hilliard equations with nonconstant mobility, Nonlinearity.20 (2007), 2365–2387. doi:10.1088/0951-7715/20/10/006.
29.
G.Schimperna and S.Zelik, Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations, Trans. Amer. Math. Soc.365 (2013), 3799–3829. doi:10.1090/S0002-9947-2012-05824-7.
30.
J.Simon, Compact sets in the space , Ann. Mat. Pura. Appl.146(1) (1986), 65–96. doi:10.1007/BF01762360.
31.
J.X.Yin, On the existence of nonnegative continuous solutions of the Cahn–Hilliard equation, J. Diff. Eqns.97 (1992), 310–327. doi:10.1016/0022-0396(92)90075-X.
