Slow motion for the Cahn-Hilliard equation in one space dimension

Alikakos, Nicholas D.; Bates, Peter W.; Fusco, Giorgio

doi:10.1016/0022-0396(91)90163-4

Cited by 184 publications

(194 citation statements)

References 24 publications

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“…See, for example, Carr, Gurtin and Slemrod 10 and Alikakos and McKinney. 16 Combining these, we get the following Proposition 2.8: Assume that W satisfies (A1Ј), (A2Ј), (A3Ј), (A4Ј) and (A5Ј), and that m (z Ϫ ,z ϩ ). There exist constants C and c 0 such that, if lрc 0 and рc 0 l ͉log l͉ , then ͉u គ ͑ x ͒Ϫq l ͑ x ͉͒рC l 2 , and l ͉u គ x ͑ x ͒Ϫq l Ј͑x͉͒рC l 2 , for x (Ϫ1/2,1/2), and, about the L 2 -estimates,…”

Section: ͑210͒mentioning

confidence: 98%

Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term

Ohnishi

Nishiura

Imai

et al. 1999

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We are primarily concerned with the variational problem with long-range interaction. This functional represents the Gibbs free energy of the microphase separation of diblock copolymer melts. The critical points of this variational problem can be regarded as the thermodynamic equilibrium state of the phase separation phenomenon. Experimentally it is well-known in the diblock copolymer problem that the final equilibrium state prefers periodic structures such as lamellar, column, spherical, double-diamond geometries and so on. We are interested in the characterization of the periodic structure of the global minimizer of the functional ͑corresponding to the strong segregation limit͒. In this paper we completely determine the principal part of the asymptotic expansion of the period with respect to ͑interfacial thickness͒, namely, we estimate the higher order error term of the period with respect to in a mathematically rigorous way in one space dimension. Moreover, we decide clearly the dependency of the constant of proportion upon the ratio of the length of two homopolymers and upon the quench depth. In the last section, we study the time evolution of the system. We first study the linear stability of spatially homogeneous steady state and derive the most unstable wavelength, if it is unstable. This is related to spinodal decomposition. Then, we numerically investigate the time evolution equation ͑the gradient flow of the free energy͒, and see that the free energy has many local minimizers and the system have some kind of sensitivity about initial data. © 1999 American Institute of Physics. ͓S1054-1500͑99͒02102-3͔In this paper we are mainly interested in characterizing the spatial periodic lamellar structure of a diblock copolymer. For the strong segregation limit, we completely determine the principal part of the asymptotic expansion of the period with respect to the interfacial thickness, which is a very small parameter, and give a mathematically rigorous proof. We also rigorously calculate the constant of proportionality and compute the dependence upon the ratio of two homopolymers, the quench depth and the long-range interaction. We consider this the first exact result for the problem. Second, we present some numerical experiments of the model, which is a certain discretization of the Cahn-Hilliard type partial differential equation with long-range interaction. We give a comparison between the rigorous result and the numerical simulation about the periodic structure of the strong segregation limit. Also, we show that a weak segregation limit on the numerical simulation has the finer period than the strong one, and the Gibbs energy functional has many other local minimizers, all of which correspond to stable steady states. This demonstrates the richness of the spatial structures of the diblock copolymer melts.

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Section: ͑210͒mentioning

confidence: 98%

Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term

Ohnishi

Nishiura

Imai

et al. 1999

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…A related analysis for stripe and ring-type solutions to the GS model is given in [23]. For the low feed-rate regime A = O (1), in §3 we analyze the stability of symmetric k-spike equilibrium solutions with respect to the small eigenvalues of The smooth curve for each symmetric solution branch has three portions with different stability properties. The curves with widely spaced dots are unstable for τ 0.…”

Section: In the Intermediate Regime O(1)mentioning

confidence: 99%

“…Similar problems involving only critical spectra occur in certain phase separation models with an associated variational principle including the Cahn-Hilliard, Allen-Cahn, and phase-field models (cf. [1]- [4]), and the analysis of lamellar states for Diblock copolymers (cf. [40]).…”

Section: In the Intermediate Regime O(1)mentioning

confidence: 99%

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Kolokolnikov

Ward

Wei

2006

Interfaces Free Bound.

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Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray-Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k−1 small eigenvalues, governing the translational stability of a symmetric kspike pattern, simultaneously cross through zero at precisely the same parameter value at which k −1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB . . . BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as ε → 0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.

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“…To name just some of these results, we recall here [5], a pioneering article in the study of slow dynamics for viscous scalar conservation law; starting from this, there are several papers concerning slow motion of internal shock layer for viscous conservation law, see for example [6], [7], [10]. Slow dynamics of interfaces has been examined also for Burgers type equation (see [2] and [12] ), for relaxation system, with the contribution [11], and for phase transition problem described by the Allen-Cahn and Cahn-Hilliard equation in [1,3,4,9].…”

Section: Introductionmentioning

confidence: 99%

Metastability for scalar conservation laws in a bounded domain

Strani

2014

ESAIM: ProcS

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Abstract. The initial-boundary-value problem for a viscous scalar conservation law in a bounded interval I = (− , ) is considered, with emphasis on the metastable dynamics, whereby the time-dependent solution develops internal transition layers that approach their steady state in an asymptotically exponentially long time interval as the viscosity coefficient ε > 0 goes to zero. We describe such behavior by deriving an ODE for the position ξ of the internal interface. The main tool of our analysis is the construction of a one-parameter family of approximate stationary solutions {U ε (·; ξ)} ξ∈I , parametrized by the location of the shock layer ξ, to be considered as an approximate invariant manifold for the problem. By using the properties of the linearized operator at U ε , we estimate the size of the layer location.

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Slow motion for the Cahn-Hilliard equation in one space dimension

Cited by 184 publications

References 24 publications

Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term

Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Metastability for scalar conservation laws in a bounded domain

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