Singularly perturbed 1D Cahn–Hilliard equation revisited

Bonfoh, Ahmed; Grasselli, Maurizio; Miranville, Alain

doi:10.1007/s00030-010-0075-0

Cited by 25 publications

(12 citation statements)

References 44 publications

(76 reference statements)

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“…The case ǫ > 0 and α = 0 is a very challenging equation (see [28][29][30]44], cf. also [4,21,52,53] for the 1D case) which becomes much nicer in presence of viscosity (cf. [2,3,5,22,34]) In particular, in the latter case, solutions regularize in finite time.…”

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confidence: 99%

Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions

Cavaterra¹,

Grasselli²,

Wu³

2014

Communications on Pure &Amp; Applied Analysis

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We consider a non-isothermal modified Cahn-Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation. The resulting system was studied in the case of no-flux boundary conditions. Here we analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with bounded energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz-Simon inequality.The Cahn-Hilliard equation is a cornerstone in Materials Science since it gives a fairly good description of phase separation processes in binary alloys (see, e.g., [7,40,41] and references therein). The early stage of such a phenomenon is called spinodal decomposition. A modification of the Cahn-Hilliard equation has been proposed in [18] to account for rapid spinodal decomposition in certain materials (see also [19,20]). This modified equation reads as followswhere ε > 0 is a relaxation time, χ represents the (relative) concentration of one component and µ is the so-called chemical potential given by µ = −∆χ + αχ t + f (χ).

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confidence: 99%

Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions

Cavaterra¹,

Grasselli²,

Wu³

2014

Communications on Pure &Amp; Applied Analysis

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“…Assuming κ = q = δ = 0, (1.5) reads 6) known as the Cahn-Hilliard equation [8,9,50,51]. It describes the process of spinodal decomposition.…”

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confidence: 99%

Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects

Coclite

Ruvo

2021

Z. Angew. Math. Phys.

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The Kuramto–Sivashinsky equation with anisotropy effects models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. Written in terms of the step slope, it can be represented in a form similar to a convective Cahn–Hilliard equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

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“…However, in the aforementioned study, 9 the convergence of the exponential attractors is obtained with respect to a norm that depends on . In another literature, 12 the authors construct a family of exponential attractors, which is continuous with respect to by using a metric that does not depend on as goes to zero. The aforementioned bibliography is confined to one-dimensional model; there is vast literature of works about Equation 1.1 in more that one spatial dimension (among others, see previous works [13][14][15][16][17] ).…”

Section: Hyperbolic Relaxation Of Cahn-hilliard Equationmentioning

confidence: 99%

Slow dynamics for the hyperbolic Cahn‐Hilliard equation in one‐space dimension

Folino

Lattanzio

2019

Math Methods in App Sciences

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The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn‐Hilliard equation in one‐space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an “energy approach," already proposed for various evolution PDEs, including the Allen‐Cahn and the Cahn‐Hilliard equations. In particular, we shall prove that certain solutions maintain a N‐transition layer structure for a very long time, thus proving their metastable dynamics. More precisely, we will show that, for an exponentially long time, such solutions are very close to piecewise constant functions assuming only the minimal points of the potential, with a finitely number of transition layers, which move with an exponentially small velocity.

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Singularly perturbed 1D Cahn–Hilliard equation revisited

Cited by 25 publications

References 44 publications

Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions

Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions

Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects

Slow dynamics for the hyperbolic Cahn‐Hilliard equation in one‐space dimension

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