Relaxation to Equilibrium in the One-Dimensional Cahn--Hilliard Equation

2014 ◽  
Vol 46 (1) ◽  
pp. 720-756 ◽  
Author(s):  
Felix Otto ◽  
Maria G. Westdickenberg
Keyword(s):  
One Dimensional ◽  
Hilliard Equation ◽  
Relaxation To Equilibrium ◽  
Cahn Hilliard Equation ◽  
The One

Steady states of the one-dimensional Cahn–Hilliard equation

1993 ◽  
Vol 123 (6) ◽  
pp. 1071-1098 ◽  
Author(s):  
A. Novick-Cohen ◽  
L. A. Peletier
Keyword(s):  
Steady States ◽  
Interval Length ◽  
One Dimensional ◽  
Average Mass ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
The One ◽  
Secondary Bifurcations ◽  
Steady State Solutions ◽  
Spinodal Region

SynopsisThe steady states of the Cahn–Hilliard equation are studied as a function of interval length, L, and average mass, m. We count the number of nontrivial monotone increasing steady state solutions and demonstrate that if m lies within the spinodal region then for a.e. there is an even number of such solutions and for a.e. there is an odd number of such solutions. If m lies within the metastable region, then for a.e. L > 0 there is an even number of solutions. Furthermore, we prove that for all values of m, there are no secondary bifurcations.

Counting stationary solutions of the Cahn–Hilliard equation by transversality arguments

1995 ◽  
Vol 125 (2) ◽  
pp. 351-370 ◽  
Author(s):  
M. Grinfeld ◽  
A. Novick-Cohen
Keyword(s):  
Stationary Solutions ◽  
Cubic Nonlinearity ◽  
One Dimensional ◽  
Level Curves ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
The One

In this paper we use arguments based on Picard-Fuchs equations and transversality of intersections of level curves to obtain an exact count of the number of stationary solutions of the one-dimensional Cahn-Hilliard equation with a cubic nonlinearity.

scholarly journals Convergence of the One-Dimensional Cahn--Hilliard Equation

2012 ◽  
Vol 44 (5) ◽  
pp. 3458-3480 ◽  
Author(s):  
Giovanni Bellettini ◽  
Lorenzo Bertini ◽  
Mauro Mariani ◽  
Matteo Novaga
Keyword(s):  
One Dimensional ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
The One
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