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

2011 ◽  
pp. 327-339 ◽  
Author(s):  
Carina Geldhauser ◽  
Matteo Novaga
Keyword(s):  
One Dimensional ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Semidiscrete Scheme

Periodic structures in a van der Waals fluid

1998 ◽  
Vol 128 (2) ◽  
pp. 235-250 ◽  
Author(s):  
Paul C. Fife ◽  
Xiao-Ping Wang
Keyword(s):  
Periodic Structures ◽  
Layer Structure ◽  
Travelling Waves ◽  
Van Der Waals ◽  
Average Density ◽  
One Dimensional ◽  
Van Der Waals Fluid ◽  
Hilliard Equation ◽  
Density Relation ◽  
Cahn Hilliard Equation

A system of partial differential equations modelling a van der Waals fluid or an elastic medium with nonmonotone pressure-density relation is studied. As the system changes type, regularisations are considered. The existence of one-dimensional periodic travelling waves, with prescribed average density in a certain range, average velocity and wavelength, is proved. They exhibit layer structure when the regularisation parameter is small. Similarities with the Cahn–Hilliard equation are explored.

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.

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