Global bifurcation and stable two-phase separation for a phase field model in a disk

Yasuhito Miyamoto

doi:10.3934/dcds.2011.30.791

Article Contents

2011, Volume 30, Issue 3: 791-806. Doi: 10.3934/dcds.2011.30.791

This issue Previous Article Pullback attractors for globally modified Navier-Stokes equations with infinite delays Next Article Coherent lists and chaotic sets

Global bifurcation and stable two-phase separation for a phase field model in a disk

Yasuhito Miyamoto^1,

1.
Department of Mathematics, Tokyo Institute of Technology, O-Okayama, Meguro-ku, Tokyo 152-8551

Received: January 2010

Revised: October 2010

Published: September 2011

Abstract / Introduction Related Papers Cited by

Abstract

Let D:$=\{(x,y);\ x^2+y^2 < l^2\}\subset\R^2$. We study the shape of a local minimizer of the problem

$E(u)=\int_{D}(\frac{|\nabla u|^2}{2}+\lambda W(u)) dx\ \ $subject to$ \ \ m=\frac{1}{|D|}\int_Dudx $

and study the global structure of critical points. We show that for an arbitrary potential $W\in C^4$ every level set of every nonconstant local minimizer is a $C^1$-curve and it divides $D$ into exactly two simply connected subdomains. Next we consider the case $W(u)=(u^2-1)^2/4$ (Cahn-Hilliard equation). When $\lambda$ varies and $m$ is fixed, we show that this problem has an unbounded continuum of critical points. When $m$ varies and $\lambda$ is fixed, we show that this problem has a bounded continuum meeting at two different points on the trivial branch. Moreover, we show that in each case a bifurcating critical point is stable (a local minimizer) near the bifurcation point in a certain parameter range. The main technique is the nodal curve which relates the shape with the Morse index. We do not use a small parameter or the $\Gamma$-convergence technique.

Keywords:

Phase separation; Cahn-Hilliard equation; Local minimizer; Second eigenvalue; Nodal curve.

Mathematics Subject Classification: Primary: 35B32, 35B36; Secondary: 35J20, 35J61.

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