A semidiscrete scheme for a one-dimensional Cahn–Hilliard equation

Geldhauser, Carina; Novaga, Matteo

doi:10.4171/ifb/260

Cited by 7 publications

(6 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case of hysteretic interface motion, there seems to be no rigorous result -neither for the lattice nor the viscous approximation -that derives (1.9) and (1.10) rigorously from the dynamics for ε > 0. Previous results for the lattices (1.1) or (1.2) are either restricted to standing interfaces, see [GN11] and [BGN13] for type-I and type-II interfaces, respectively, or do not capture the dynamics of moving interfaces completely, e. g. [BNP06].…”

Section: A the Discrete Heat Kernel 1 Introductionmentioning

confidence: 99%

Interface Dynamics in Discrete Forward-Backward Diffusion Equations

Helmers¹,

Herrmann²

2013

Multiscale Model. Simul.

View full text Add to dashboard Cite

We study the motion of phase interfaces in a diffusive lattice equation with bistable nonlinearity and derive a free boundary problem with hysteresis to describe the macroscopic evolution in the parabolic scaling limit.The first part of the paper deals with general bistable nonlinearities and is restricted to numerical experiments and heuristic arguments. We discuss the formation of macroscopic data and present numerical evidence for pinning, depinning, and annihilation of interfaces. Afterwards we identify a generalized Stefan condition along with a hysteretic flow rule that characterize the dynamics of both standing and moving interfaces.In the second part, we rigorously justify the limit dynamics for single-interface data and a special piecewise affine nonlinearity. We prove persistence of such data, derive upper bounds for the macroscopic interface speed, and show that the macroscopic limit can indeed be described by the free boundary problem. The fundamental ingredient to our proofs is a representation formula that links the solutions of the nonlinear lattice to the discrete heat kernel and enables us to derive macroscopic compactness results in the space of continuous functions. Keywords:forward-backward diffusion in lattices, coarse graining for gradient flows, hysteretic models for phase transitions, pinning and depinning of interfaces, regularization of ill-posed parabolic PDEs MSC (2010):

show abstract

Section: A the Discrete Heat Kernel 1 Introductionmentioning

confidence: 99%

Interface Dynamics in Discrete Forward-Backward Diffusion Equations

Helmers¹,

Herrmann²

2013

Multiscale Model. Simul.

View full text Add to dashboard Cite

show abstract

“…We observe that equation (1.1) is not the only way to regularize the ill-posed gradient flow equation of the functional (1.4): other regularizations have been considered in the literature, see for instance [15,7,10,9,19]. In particular, in [7] it is proposed an implicit variational scheme for the functional (1.4) which converges to (1.6) as the discretization parameter tends to zero.…”

Section: Introductionmentioning

confidence: 99%

Convergence of the One-Dimensional Cahn--Hilliard Equation

Bellettini

Bertini

Mariani

et al. 2012

SIAM J. Math. Anal.

Self Cite

View full text Add to dashboard Cite

We consider the Cahn-Hilliard equation in one space dimension with scaling parameter epsilon, i.e., u(t) = (W'(u) - epsilon(2)u(xx))(xx), where W is a nonconvex potential. In the limit epsilon down arrow 0, under the assumption that the initial data are energetically well prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard equation

show abstract

“…Moreover, for initial data with p j (0) < p * for all j ∈ Z no u j can reach the spinodal region from below, and if p j (0) ∈ [p * , p * ] then no interface moves at all. See Geldhauser and Novaga (2011) or Helmers and Herrmann (2013) for more details.…”

Section: Microscopic Interface Dynamicsmentioning

confidence: 99%

“…The case of standing interfaces is much simpler and has been settled in Geldhauser and Novaga (2011) by localizing standard methods for parabolic PDEs.…”

Section: Rigorous Justification Of the Limitmentioning

confidence: 99%

A free boundary problem for interfaces in forward-backward lattice diffusion

Helmers

2015

IFAC-PapersOnLine

View full text Add to dashboard Cite

We examine the motion of phase interfaces in a diffusive lattice equation with bistable nonlinearity. We first show by heuristic arguments that the macroscopic evolution in the parabolic scaling limit is governed by a free boundary problem with hysteresis. Then, starting out from results for a piecewise affine nonlinearity whose backward region is a single point, we discuss ideas for a rigorous proof.

show abstract

A semidiscrete scheme for a one-dimensional Cahn–Hilliard equation

Cited by 7 publications

References 14 publications

Interface Dynamics in Discrete Forward-Backward Diffusion Equations

Interface Dynamics in Discrete Forward-Backward Diffusion Equations

Convergence of the One-Dimensional Cahn--Hilliard Equation

A free boundary problem for interfaces in forward-backward lattice diffusion

Contact Info

Product

Resources

About