zbMATH — the first resource for mathematics

Slow motion for the Cahn-Hilliard equation in one space dimension. (English) Zbl 0753.35042
The authors consider the Cahn-Hilliard equation in one space dimension \[ u_ t=(-\varepsilon^ 2 u_{xx}+W'(u))_{xx}\qquad \text{on}\qquad (0,1)\times(0,\infty),\tag{\text{CH}} \]
\[ u_ x=u_{xx}=0,\quad x=0,1,\quad t>0;\qquad u(x,0)=u_ 0(x). \] Hypothesis: \(W(u)\) is a \(C^ 4\) function with exactly three critical points, \(\alpha<\beta<\gamma\), with \(\alpha,\beta\) local minima and \(\gamma\) a local maximum and \(W\geq 0\), \(W''(\alpha)\), \(W''(\beta)\), \(-W''(\gamma)>0\).
The authors seek the invariant manifold \({\mathcal M}\) for (CH) under the above hypothesis, i.e. seek the solution \(u(x,t)=\hat u(\xi(t),x)\), where \(\hat u(\xi,x)\) and \(\xi(t)\) satisfy \(\dot\xi(t)=b(\xi)\) and \(- \varepsilon^ 2\hat u_{xxxx}+(W'(\hat u_ x))_ x=b(\xi)\hat u\).
They prove the Theorem: There exists \(\varepsilon_ 0>0\) such that for \(\varepsilon<\varepsilon_ 0\) there are functions \(\hat u(\xi,\cdot)\in H^{4\alpha}\) and \(b(\xi)\) defined in \((-\ell+\delta,\ell-\delta)\), with the following properties:
(i) \(\xi\to\hat u(\xi,\cdot)\), \(\xi\to b(\xi)\) are Lipschitz continuous, (ii) \({\mathcal M}=\{u=\hat u(\xi,\cdot);\;\xi\in(-\ell+\delta,\ell- \delta)\}\) is an invariant manifold for the dynamical system defined by (CH) in \(H^{4\alpha}\), (iii) The flow on \({\mathcal M}\) is described by the ordinary differential equation \(\dot\xi=b(\xi)\). (iv) The following estimates hold: \(\| \hat u-u(\xi,\cdot)\|_{H^ 2}=O(\exp(- \nu\delta_ \xi/\varepsilon))\) and \(b(\xi)=O(\exp(-2\nu\delta_ \xi/\varepsilon))\) and, \(b(0)=0\), where \(u(\xi,x)=u(x-\xi)\), \(\delta_ \xi=\ell-|\xi|\), and \(u(x)\) is the equilibrium (time independent solution) of (CH).

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
58J70 Invariance and symmetry properties for PDEs on manifolds
37C10 Dynamics induced by flows and semiflows
Full Text: DOI
[1] Alexiades, V.; Aifantis, E.C., On the thermodynamic theory of fluid interfaces, infinite intervals, equilibrium solutions, and minimizers, J. colloid. interface science, 111, 119-132, (1986)
[2] Amann, H., Periodic solutions of semilinear parabolic equations, (), 1-29
[3] Alikakos, N.D.; McKinney, W.R., Remarks on the equilibrium theory for the Cahn-Hilliard equation in one space dimension, (), 75-93
[4] Angenent, S.B., The Morse-Smale property for a semilinear parabolic equation, J. differential equations, 62, 427-442, (1986) · Zbl 0581.58026
[5] Bates, P.W.; Fife, P.C., Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Phys. D, 43, 335-348, (1990) · Zbl 0706.58074
[6] Bronsard, L., ()
[7] {\scB. Bronsard and R. V. Kohn}, On the slowness of phase boundary motion in one space dimension, CPAM, to appear. · Zbl 0761.35044
[8] Brunovsky, P.; Fiedler, B.; Brunovsky, P.; Fiedler, B., (II) connecting orbits in scalar reaction-diffusion equations II: the complete solution, Dynamics reported, Dynamics reported, 1, 57-89, (1988), preprint · Zbl 0679.35047
[9] Cahn, J.W., On spinodal decomposition, Acta metallurgica, 9, 795-801, (1961)
[10] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial free energy, J. chem. phys., 28, 258-267, (1958)
[11] Carr, J.; Gurtin, M.; Slemrod, M., Structured phase transitions on a finite interval, Arch. rational mech. anal., 86, 317-351, (1984) · Zbl 0564.76075
[12] Carr, J.; Pego, R.L., Metastable patterns in solutions of \(ut = ε\^{}\{2\}uxx−ƒ(u)\), Cpam, 42, 523-576, (1989) · Zbl 0685.35054
[13] deMottoni, P.; Schatzman, M., Evolution géométrique d’interfaces, C.R. acad. sci. Paris, (1988)
[14] Elliott, C.M.; French, D.A., Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. appl. math., 38, 97-128, (1987) · Zbl 0632.65113
[15] Elliott, C.M.; Zheng, S., On the Cahn-Hilliard equation, Arch. rational mech. anal., 96, 339-357, (1986) · Zbl 0624.35048
[16] {\scL. C. Evans and P. E. Souganidis}, A PDE approach to geometric optics for certain semilinear parabolic equations, preprint. · Zbl 0692.35014
[17] {\scP. C. Fife}, personal communication.
[18] Fife, P.C.; Gill, G.S., The phase-field description of mushy zones, Phys. D, 35, 267-275, (1989)
[19] Friedlin, M.I.; Friedlin, M.I., (II) geometric optics approach to reaction-diffusion equations, Ann. probab., SIAM J. appl. math., 46, 222-232, (1986) · Zbl 0626.35047
[20] {\scG. Fusco}, A geometric approach to the dynamics of \(ut = ε\^{}\{2\}uxx + ƒ(u)\) for small ε, in “Proceedings of the Stuttgart conference in honor of J. K. Hale” (K. Kirchgässner, Ed.). · Zbl 0715.35038
[21] {\scG. Fusco}, unpublished.
[22] Fusco, G.; Hale, J.K., Slow motion manifolds, dormant instability and singular perturbations, Dynamics differential equations, 1, 75-94, (1989) · Zbl 0684.34055
[23] Gunton, J.D.; Droz, M., Introduction to the theory of metastable and unstable states, (), No. 183
[24] Gurtin, M.E., Some results and conjectures in the gradient theory of phase transitions, () · Zbl 0579.73016
[25] Gurtin, M.; Matano, H., On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. appl. math., 46, 301-317, (1988) · Zbl 0665.76120
[26] Henry, D., Geometric theory of semilinear parabolic equations, (), No. 840 · Zbl 0456.35001
[27] Henry, D., Some infinite dimensional Morse-Smale systems defined by parabolic equations, J. differential equations, 59, 165-205, (1985) · Zbl 0572.58012
[28] {\scJ. B. Keller, J. Rubinstein and P. Sternberg}, Fast reaction, slow diffusion, and curve shortening, preprint. · Zbl 0701.35012
[29] Langer, J.S., Theory of spinodal decomposition in alloys, Ann. physics, 65, 53-86, (1971)
[30] Luckhaus, S.; Modica, L., The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. rational mech. anal., 107, 71-83, (1989) · Zbl 0681.49012
[31] Modica, L., Gradient theory of phase transitions and singular perturbation, Arch. rational mech. anal., 98, 123-142, (1986)
[32] McKinney, W., ()
[33] {\scJ. Neu}, unpublished lecture notes.
[34] Novick-Cohen, A., The nonlinear Cahn-Hilliard equation: transition from spinodal decomposition to nucleation behavior, J. statist. phys., 38, 707-723, (1985)
[35] Novick-Cohen, A.; Segal, L.A., Nonlinear aspects of the Cahn-Hilliard equation, Phys. D, 10, 277-298, (1984)
[36] Nishiura, Y.; Fuji, H., SLEP method and the stability of singularly perturbed solutions with multiple internal transition layers in reaction-diffusion systems, (), 211-230
[37] {\scR. L. Pego}, Front migration in the nonlinear Cahn-Hilliard equation, preprint. · Zbl 0701.35159
[38] Sternberg, P., The effect of a singular perturbation on nonconvex variational problems, Arch. rational mech. anal., 101, 209-260, (1988) · Zbl 0647.49021
[39] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, () · Zbl 0871.35001
[40] Weinberger, H.F., On metastable patterns in parabolic systems, Rend. accad. naz. lincei, 77, 291-313, (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.