Homogenization of Cahn–Hilliard-type equations via evolutionary $$\varvec{\Gamma }$$-convergence

Liero, Matthias; Reichelt, Sina

doi:10.1007/s00030-018-0495-9

Cited by 14 publications

(21 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since y ε (t) → y(t) strongly in Y and y(t) is shift-invariant, it follows that T ε y ε (t) → y(t) strongly in Y . As a result of this and using the weak convergence in (27), similarly to the proof of (32), we obtain that lim inf ε→0 E ε (y ε (t)) ≥ E hom (y(t)) (by employing Proposition 3.5). As a consequence, using the additional assumption E ε (y ε (0)) → E hom (y(0)), we obtain lim sup DR hom (ẏ(s)),ẏ(s) Y * 0 ,Y0 ds.…”

Section: By the Density Ofsupporting

confidence: 53%

“…The notion of periodic two-scale convergence [39,2] (see also [28]) and the periodic unfolding procedure [13] (see also [14,50,34]) are prominent and useful tools in multiscale modeling and homogenization suited for problems involving periodic coefficients. We refer to some of the many problems treated using these methods [28,12,21,34,35,32,27,22]. In the stochastic setting, the notion of two-scale convergence is generalized in [9] (see also [4,45]) and in [54] (see also [29,19,23]).…”

mentioning

confidence: 99%

“…Related results. In the periodic setting homogenization results of this type are obtained for quasilinear parabolic equations, e.g., in [38,51,20] (via two-scale convergence and unfolding), for reaction-diffusion systems with different diffusion length scales in [32] (via unfolding), for Cahn-Hilliard type gradient flows in [27] (via unfolding). In the stochastic case, parabolic type equations are treated in [53,16,24,17].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Stochastic homogenization of \Lambda -convex gradient flows

Heida¹,

Neukamm²,

Varga³

2021

Discrete &Amp; Continuous Dynamical Systems - S

View full text Add to dashboard Cite

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals.

show abstract

Section: By the Density Ofsupporting

confidence: 53%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Stochastic homogenization of \Lambda -convex gradient flows

Heida¹,

Neukamm²,

Varga³

2021

Discrete &Amp; Continuous Dynamical Systems - S

View full text Add to dashboard Cite

show abstract

“…In some cases, this method facilitates a more straightforward, and operator theory flavored, analysis of periodic homogenization problems. In recent years periodic unfolding has been applied to a large variety of multiscale problems; e.g., see [11,17,37,38,34,43,10,29,42,21]. For a systematic investigation of two-scale calculus associated with the use of the periodic unfolding method we refer to [14,44,45,37,12].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Unfolding and Homogenization of Spring Network Models

Neukamm¹,

Varga²

2018

Multiscale Model. Simul.

View full text Add to dashboard Cite

The aim of our work is to provide a simple homogenization and discrete-to-continuum procedure for energy driven problems involving stochastic rapidly oscillating coefficients. Our intention is to extend the periodic unfolding method to the stochastic setting. Specifically, we recast the notion of stochastic two-scale convergence in the mean by introducing an appropriate stochastic unfolding operator. This operator admits similar properties as the periodic unfolding operator, leading to an uncomplicated method for stochastic homogenization. Second, we analyze the discrete-to-continuum (resp., stochastic homogenization) limit for a rate-independent system describing a network of linear elasto-plastic springs with random coefficients.

show abstract

“…Rigorous homogenization of two-phase emulsion with fixed geometry of (microscale) interfaces and surface tension effects has been considered in [31], [32]. Upscaled models for Cahn-Hilliard type equations have been derived in [41], [40] via the asymptotic expansion method and in [30] via the two-scale convergence approach.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of evolutionary Stokes–Cahn–Hilliard equations for two-phase porous media flow

Baňas

Mahato

2017

ASY

View full text Add to dashboard Cite

We consider homogenization of a phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects. The pore-scale model consists of a strongly coupled system of time-dependent Stokes-Cahn-Hilliard equations. In the considered model the fluids are separated by an evolving diffuse interface of a finite width, which is assumed to be independent of the scale parameter ε. We obtain upscaled equations for the considered model by a rigorous two-scale convergence approach.

show abstract

Homogenization of Cahn–Hilliard-type equations via evolutionary $$\varvec{\Gamma }$$-convergence

Cited by 14 publications

References 39 publications

Stochastic homogenization of \Lambda -convex gradient flows

Stochastic homogenization of \Lambda -convex gradient flows

Stochastic Unfolding and Homogenization of Spring Network Models

Homogenization of evolutionary Stokes–Cahn–Hilliard equations for two-phase porous media flow

Contact Info

Product

Resources

About