Domain coarsening in a subdiffusive Allen–Cahn equation

2015 ◽  
Vol 308 ◽  
pp. 52-58 ◽  
Author(s):  
M. Abu Hamed ◽  
A.A. Nepomnyashchy
Keyword(s):  
Cahn Equation ◽  
Domain Coarsening

Domain coarsening by grain boundary migration in ordered Ni4Mo

1986 ◽  
Vol 44 ◽  
pp. 560-561
Author(s):  
K. Vasudevan ◽  
H. P. Kao ◽  
C. R. Brooks ◽  
E. E. Stansbury
Keyword(s):  
Grain Boundary ◽  
Grain Boundaries ◽  
Short Range ◽  
Boundary Migration ◽  
Grain Boundary Migration ◽  
Rapid Cooling ◽  
Temperature Range ◽  
Fee Structure ◽  
Domain Coarsening ◽  
Boundary Reaction

The Ni4Mo alloy has a short-range ordered fee structure (α) above 868°C, but transforms below this temperature to an ordered bet structure (β) by rearrangement of atoms on the fee lattice. The disordered α, retained by rapid cooling, can be ordered by appropriate aging below 868°C. Initially, very fine β domains in six different but crystallographically related variants form and grow in size on further aging. However, in the temperature range 600-775°C, a coarsening reaction begins at the former α grain boundaries and the alloy also coarsens by this mechanism. The purpose of this paper is to report on TEM observations showing the characteristics of this grain boundary reaction.

scholarly journals Mass-conserving diffusion-based dynamics on graphs

2021 ◽  
pp. 1-49
Author(s):  
J.M BUDD ◽  
Y. VAN GENNIP
Keyword(s):  
Multiscale Modeling ◽  
Classification Problems ◽  
Theoretical Justification ◽  
Finite Graphs ◽  
Discrete Scheme ◽  
Cahn Equation ◽  
Separating Flows ◽  
Special Case ◽  
Appl Math ◽  
Convergence Of The Scheme

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

scholarly journals BSDEs with polynomial growth generators

2000 ◽  
Vol 13 (3) ◽  
pp. 207-238 ◽  
Author(s):  
Philippe Briand ◽  
René Carmona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Polynomial Growth ◽  
Probabilistic Representation ◽  
Existence And Uniqueness Results ◽  
State Variable ◽  
Cahn Equation ◽  
Uniqueness Results ◽  
Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

Late Stage Domain Coarsening Dynamics of Lamellar Block Copolymers

2021 ◽  
pp. 727-731
Author(s):  
Maninderjeet Singh ◽  
Wenjie Wu ◽  
Vinay Nuka ◽  
Joseph Strzalka ◽  
Jack F. Douglas ◽  
...  
Keyword(s):  
Block Copolymers ◽  
Late Stage ◽  
Domain Coarsening ◽  
Coarsening Dynamics

scholarly journals Multifaceted phase ordering kinetics of an antiferromagnetic spin-1 condensate

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Joanna Pietraszewicz ◽  
Aleksandra Seweryn ◽  
Emilia Witkowska
Keyword(s):  
Scaling Laws ◽  
Study Phase ◽  
One Dimensional ◽  
Phase Domain ◽  
Ordering Kinetics ◽  
Domain Coarsening ◽  
Long Time ◽  
Phase Ordering ◽  
Kinetics Of ◽  
Antiferromagnetic Spin

AbstractWe study phase domain coarsening in the long time limit after a quench of magnetic field in a quasi one-dimensional spin-1 antiferromagnetic condensate. We observe that the growth of correlation length obeys scaling laws predicted by the two different models of phase ordering kinetics, namely the binary mixture and vector field. We derive regimes of clear realization for both of them. We demonstrate appearance of atypical scaling laws, which emerge in intermediate regions.

Geometrical image segmentation by the Allen–Cahn equation

2004 ◽  
Vol 51 (2-3) ◽  
pp. 187-205 ◽  
Author(s):  
Michal Beneš ◽  
Vladimı́r Chalupecký ◽  
Karol Mikula
Keyword(s):  
Image Segmentation ◽  
Cahn Equation
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