Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory

@article{Hutchinson2000ConvergenceOP,
  title={Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory},
  author={J. Hutchinson and Y. Tonegawa},
  journal={Calculus of Variations and Partial Differential Equations},
  year={2000},
  volume={10},
  pages={49-84}
}
Abstract. We study the general asymptotic behavior of critical points, including those of non-minimal energy type, of the functional for the van der Waals-Cahn-Hilliard theory of phase transitions. We prove that the interface is close to a hypersurface with mean curvature zero when no Lagrange multiplier is present, and with locally constant mean curvature in general. The energy density of the limiting measure has integer multiplicity almost everywhere modulo division by a surface energy… Expand
Stable phase interfaces in the van der Waals–Cahn–Hilliard theory
Abstract We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals–Cahn–Hilliard energyExpand
Convergence of phase-field approximations to the Gibbs–Thomson law
We prove the convergence of phase-field approximations of the Gibbs–Thomson law. This establishes a relation between the first variation of the Van der Waals–Cahn–Hilliard energy and the firstExpand
On the van der Waals–Cahn–Hilliard phase-field model and its equilibria conditions in the sharp interface limit
  • W. Dreyer, C. Kraus
  • Chemistry
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2010
We study the thermodynamic consistency of phase-field models, which include gradient terms of the density ρ in the free-energy functional such as the van der Waals–Cahn–Hilliard model. It is wellExpand
Mathematik in den Naturwissenschaften Leipzig Convergence of phase-field approximations to the Gibbs-Thomson law by
We prove the convergence of phase-field approximations of the Gibbs–Thomson law. This establishes a relation between the first variation of the Van-der-Waals–Cahn–Hilliard energy and the firstExpand
The sharp interface limit of the van der Waals-Cahn-Hilliard phase model for fixed and time dependent domains
We first study the thermodynamic consistency of phase field models which include gradient terms of the density ρ in the free energy function, such as the van der Waals–Cahn–Hilliard model. It isExpand
Multiple solutions for the Van der Waals–Allen–Cahn–Hilliard equation with a volume constraint
We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals–Allen–Cahn–Hilliard type, satisfying a linear volumeExpand
Phase field model with a variable chemical potential
  • Y. Tonegawa
  • Chemistry, Physics
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2002
We study some asymptotic behaviour of phase interfaces with variable chemical potential under the uniform energy bound. The problem is motivated by the Cahn-Hilliard equation, where one has a controlExpand
A diffused interface with the advection term in a Sobolev space
We study the asymptotic limit of diffused surface energy in the van der Waals--Cahn--Hillard theory when an advection term is added and the energy is uniformly bounded. We prove that the limitExpand
A Higher Order Asymptotic Problem Related to Phase Transitions
  • R. Moser
  • Computer Science, Mathematics
  • SIAM J. Math. Anal.
  • 2005
TLDR
An asymptotic analysis of a family of functionals used in the van der Waals--Cahn--Hilliard theory of phase transitions gives rise to a generalized area functional in the limit, which is a generalization of the Willmore functional. Expand
A phase field formulation of the Willmore problem
In this paper, we demonstrate, through asymptotic expansions, the convergence of a phase field formulation to model surfaces minimizing the mean curvature energy with volume and surface areaExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 46 REFERENCES
Gradient theory of phase transitions with boundary contact energy
Abstract We study the asymptotic behavior as e → 0+ of solutions of the variational problems for the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid. We assume that the internalExpand
The gradient theory of phase transitions and the minimal interface criterion
In this paper I prove some conjectures of GURTIN [15] concerning the Van der Waals-Cahn-Hilliard theory of phase transitions. Consider a fluid, under isothermal conditions and confined to a boundedExpand
The Gibbs-Thompson relation within the gradient theory of phase transitions
This paper discusses the asymptotic behavior as ɛ → 0+ of the chemical potentials λɛ associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phaseExpand
Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids
Abstract In this paper we extend the Van der Waals-Cahn-Hilliard theory of phase transitions to the case of a mixture of n non-interacting fluids. By describing the state of the mixture as given by aExpand
Phase Transitions and Generalized Motion by Mean Curvature
We study the limiting behavior of solutions to appropriately rescaled versions of the Allen-Cahn equation, a simplified model for dynamic phase transitions. We rigorously establish the existence inExpand
On the Existence of High Multiplicity Interfaces
A bstract . In many singularly perturbed Ginzburg–Landau type partial differential equations, such as the Allen–Cahn equation, the nonlocal Allen–Cahn equation, and the Cahn–Hilliard equation, theExpand
Ginzburg-Landau equation and motion by mean curvature, II: Development of the initial interface
In this paper, we study the short time behavior of the solutions of a sequence of Ginzburg-Landau equations indexed by ∈. We prove that under appropriate assumptions on the initial data, solutionsExpand
A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening
Abstract A microscopic diffusional theory for the motion of a curved antiphase boundary is presented. The interfacial velocity is found to be linearly proportional to the mean curvature of theExpand
Front migration in the nonlinear Cahn-Hilliard equation
  • R. Pego
  • Mathematics
  • Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1989
The method of matched asymptotic expansions is used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in N > 1 space dimensions. The expansion is formally valid whenExpand
Slow motion in the gradient theory of phase transitions via energy and spectrum
Abstract. We describe some aspects of the Cahn-Hilliard and related equations. In particular we consider the dynamics of almost spherical interfaces and establish that almost spherical interfacesExpand
...
1
2
3
4
5
...