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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).

##### MSC:
 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
##### Keywords:
melted binary alloy; invariant manifold
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##### References:
  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)  Amann, H., Periodic solutions of semilinear parabolic equations, (), 1-29  Alikakos, N.D.; McKinney, W.R., Remarks on the equilibrium theory for the Cahn-Hilliard equation in one space dimension, (), 75-93  Angenent, S.B., The Morse-Smale property for a semilinear parabolic equation, J. differential equations, 62, 427-442, (1986) · Zbl 0581.58026  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  Bronsard, L., ()  {\scB. Bronsard and R. V. Kohn}, On the slowness of phase boundary motion in one space dimension, CPAM, to appear. · Zbl 0761.35044  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  Cahn, J.W., On spinodal decomposition, Acta metallurgica, 9, 795-801, (1961)  Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial free energy, J. chem. phys., 28, 258-267, (1958)  Carr, J.; Gurtin, M.; Slemrod, M., Structured phase transitions on a finite interval, Arch. rational mech. anal., 86, 317-351, (1984) · Zbl 0564.76075  Carr, J.; Pego, R.L., Metastable patterns in solutions of $$ut = ε\^{}\{2\}uxx−ƒ(u)$$, Cpam, 42, 523-576, (1989) · Zbl 0685.35054  deMottoni, P.; Schatzman, M., Evolution géométrique d’interfaces, C.R. acad. sci. Paris, (1988)  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  Elliott, C.M.; Zheng, S., On the Cahn-Hilliard equation, Arch. rational mech. anal., 96, 339-357, (1986) · Zbl 0624.35048  {\scL. C. Evans and P. E. Souganidis}, A PDE approach to geometric optics for certain semilinear parabolic equations, preprint. · Zbl 0692.35014  {\scP. C. Fife}, personal communication.  Fife, P.C.; Gill, G.S., The phase-field description of mushy zones, Phys. D, 35, 267-275, (1989)  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  {\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  {\scG. Fusco}, unpublished.  Fusco, G.; Hale, J.K., Slow motion manifolds, dormant instability and singular perturbations, Dynamics differential equations, 1, 75-94, (1989) · Zbl 0684.34055  Gunton, J.D.; Droz, M., Introduction to the theory of metastable and unstable states, (), No. 183  Gurtin, M.E., Some results and conjectures in the gradient theory of phase transitions, () · Zbl 0579.73016  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  Henry, D., Geometric theory of semilinear parabolic equations, (), No. 840 · Zbl 0456.35001  Henry, D., Some infinite dimensional Morse-Smale systems defined by parabolic equations, J. differential equations, 59, 165-205, (1985) · Zbl 0572.58012  {\scJ. B. Keller, J. Rubinstein and P. Sternberg}, Fast reaction, slow diffusion, and curve shortening, preprint. · Zbl 0701.35012  Langer, J.S., Theory of spinodal decomposition in alloys, Ann. physics, 65, 53-86, (1971)  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  Modica, L., Gradient theory of phase transitions and singular perturbation, Arch. rational mech. anal., 98, 123-142, (1986)  McKinney, W., ()  {\scJ. Neu}, unpublished lecture notes.  Novick-Cohen, A., The nonlinear Cahn-Hilliard equation: transition from spinodal decomposition to nucleation behavior, J. statist. phys., 38, 707-723, (1985)  Novick-Cohen, A.; Segal, L.A., Nonlinear aspects of the Cahn-Hilliard equation, Phys. D, 10, 277-298, (1984)  Nishiura, Y.; Fuji, H., SLEP method and the stability of singularly perturbed solutions with multiple internal transition layers in reaction-diffusion systems, (), 211-230  {\scR. L. Pego}, Front migration in the nonlinear Cahn-Hilliard equation, preprint. · Zbl 0701.35159  Sternberg, P., The effect of a singular perturbation on nonconvex variational problems, Arch. rational mech. anal., 101, 209-260, (1988) · Zbl 0647.49021  Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, () · Zbl 0871.35001  Weinberger, H.F., On metastable patterns in parabolic systems, Rend. accad. naz. lincei, 77, 291-313, (1986)
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