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Existence and uniqueness of radially symmetric stationary points within the gradient theory of phase transitions
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In this article, Radially symmetric stationary points of the functional were studied under the Euler-Lagrange equation and the assumption of small energy, that is, for small e, and the existence of precisely two solutions for the corresponding Euler Lagrangian equation was proved.Abstract:
We study radially symmetric stationary points of the functionalwhere u denotes the density of a fluid confined to a container Ω, W(u) is the course-grain free energy and e accounts for surface energy Under the further assumption of small energy, that isfor small e, we prove existence of precisely two solutions for the corresponding Euler-Lagrange equation Each of these solutions is monotone in the radial direction and converges as e→0 to one of two possible radially symmetric single interface minimizers of E0 Our main tool is the method of matched asymptotic expansions from which we construct exact solutionsread more
Citations
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A bibliography on moving-free boundary problems for the heat-diffusion equation. The stefan and related problems
TL;DR: Tarzia et al. as discussed by the authors presented a bibliografía on moving and free boundary problems for the heatdiffusion equation, particularly regarding the Stefan and related problems, which contains 5869 titles referring to 588 scientific journals, 122 books, 88 symposia, 30 collections, 59 thesis and 247 technical reports.
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Connectivity of Phase Boundaries in Strictly Convex Domains
Peter Sternberg,Kevin Zumbrun +1 more
TL;DR: In this article, the authors consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained variational problem and show that in a strictly convex domain, stable critical point of the original variational problems have a connected, thin transition layer separating the two phases.
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Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation
Lia Bronsard,Barbara Stoth +1 more
TL;DR: In this paper, the authors studied the asymptotic behavior of radially symmetric solutions of the nonlocal nonlocal equation in a bounded spherically symmetric domain, where the initial data as well as their energy are bounded.
Journal ArticleDOI
The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor
TL;DR: In this article, a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn-Hilliard equation by explicit energy calculations is established, and strong non-degeneracy is proven away from turning points and points of bifurcation from homogeneous state and the dimension of the unstable manifold is calculated for all stationary states.
Convergence of the CahnHilliard Equation to the MullinsSekerka Problem in Spherical Symmetry
TL;DR: In this paper, it was shown that the solution of the Cahn-Hilliard equation converges to a solution of a stationary wave solution that corresponds to the potential W, provided the solutions are radially symmetric.
References
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Journal ArticleDOI
Free Energy of a Nonuniform System. I. Interfacial Free Energy
John W. Cahn,John E. Hilliard +1 more
TL;DR: In this article, it was shown that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc, and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2.
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On spinodal decomposition
TL;DR: In this article, the stability of a solid solution to all infinitesimal composition fluctuations is considered, taking surface tension and elastic energy into account, and it is found that for infinite isotropic solids, free from imperfections, the spinodal marks the limit of metastability to such fluctuations only if there is no change in molar volume with composition.
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The gradient theory of phase transitions and the minimal interface criterion
TL;DR: In this paper, the authors prove some conjectures of GURTIN concerning the Van der Waals-Cahn-Hilliard theory of phase transitions, and prove the existence of a phase transition in a fluid under isothermal conditions and confined to a bounded container.
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On the Cahn-Hilliard equation
Charles M. Elliott,Zheng Songmu +1 more
TL;DR: In this paper, a perturbation of the concentration of one of the phases and the equation of conservation of mass with the mass flux J being 0 [ 02U] or-Ox 9(u) -y~--~x2j.
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Front migration in the nonlinear Cahn-Hilliard equation
TL;DR: In this article, matched asymptotic expansions are used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in N > 1 space dimensions, when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature.