Stochastic Cahn–Hilliard equation with singular nonlinearity and reflection

Goudenège, Ludovic

doi:10.1016/j.spa.2009.06.008

Cited by 27 publications

(33 citation statements)

References 30 publications

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“…The case of non smooth ψ has been the object of much less research (see [8] and [15]). Moreover, when a non-logarithmic potential is considered, dynamics is changed and can lead to completely different behaviors (see [19] or [30]). …”

Section: Introductionmentioning

confidence: 99%

High Order Finite Element Calculations for the Cahn-Hilliard Equation

2011

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In this work, we propose a numerical method based on high degree continuous nodal elements for the Cahn-Hilliard evolution. The use of the p-version of the finite element method proves to be very efficient avoiding difficult computations or strategies like C 1 elements, adaptive mesh refinement, multi-grid resolution or isogeometric analysis. Beyond the classical benchmarks and comparisons with other existing methods, a numerical study has been carried out to investigate the influence of a polynomial approximation of the logarithmic free energy.Keywords Cahn-Hilliard · Partial differential equation · Bi-Laplacian · p-version of the finite element method · High order · Logarithm · Polynomial energy · Spinodal decomposition

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Section: Introductionmentioning

confidence: 99%

High Order Finite Element Calculations for the Cahn-Hilliard Equation

2011

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“…A first one consists in studying systems of Cahn-Hilliard equations to describe phase separation in multicomponent alloys (see [52], [81], [82], [85], [107], [108], [109] and [167]). We also mention the stochastic Cahn-Hilliard equation (also called the Cahn-Hilliard-Cook equation) which takes into account thermal fluctuations (see [59]; see also [17], [18], [20], [21], [43], [61], [64], [75], [123] and [131]). Then, an important generalization of the Cahn-Hilliard equation is the viscous Cahn-Hilliard equation which accounts for viscosity effects in the phase separation of polymer/polymer systems (see [177]; see also [8], [45], [54] and [83]).…”

Section: Cahn-hilliard Equation 565mentioning

confidence: 99%

The Cahn-Hilliard Equation with Logarithmic Potentials

2011

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Our aim in this article is to discuss recent issues related with the Cahn- Hilliard equation in phase separation with the thermodynamically relevant logarithmic potentials; in particular, we are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Neumann boundary conditions and then dynamic boundary conditions which account for the interactions with the walls in confined systems and have recently been proposed by physicists. We also present, in the case of dynamic boundary conditions, some numerical results

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“…It has been rigorously proved in [23] that a nonlinear term of the form ln u is not strong enough to ensure that solutions remain positive, and a reflection measure has to be added.…”

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“…Moreover, as in the second order case, an integration by parts formula for the invariant measure has been derived. Then, in [23], a singular nonlinearity of the form u −α or ln u has been considered. Existence and uniqueness of solutions have been obtained, and using the integration by parts formula as in [40], it has been proved that the reflection measure vanishes if and only if α ≥ 3.…”

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Stochastic Cahn–Hilliard Equation with Double Singular Nonlinearities and Two Reflections

Debussche¹,

Goudenège²

2011

SIAM J. Math. Anal.

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We consider a stochastic partial differential equation with two logarithmic nonlinearities, two reflections at 1 and −1, and a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of a maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche, Goudenège, and Zambotti, we obtain existence and uniqueness of a solution for initial conditions in the interval (−1, 1). Finally, we prove that the unique invariant measure is ergodic, and we give a result of exponential mixing.Introduction and main results. The Cahn-Hilliard-Cook equation is a model to describe phase separation in a binary alloy (see [6], [7], and [8]) in the presence of thermal fluctuations (see [11] and [27]). It takes the form

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Stochastic Cahn–Hilliard equation with singular nonlinearity and reflection

Cited by 27 publications

References 30 publications

High Order Finite Element Calculations for the Cahn-Hilliard Equation

High Order Finite Element Calculations for the Cahn-Hilliard Equation

The Cahn-Hilliard Equation with Logarithmic Potentials

Stochastic Cahn–Hilliard Equation with Double Singular Nonlinearities and Two Reflections

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