Peter Howard (Texas A&M)


Event Details

  • Date: Tuesday, Oct 23rd 2018 4:00:pm
  • Venue: 135 TMCB
  • Categories:
  • Topic: Phase-asymptotic stability of transition front solutions in Cahn-Hilliard equations and systems
  • Speaker: Peter Howard
  • College/Organization: (Texas A&M)

Phase-asymptotic stability of transition front solutions
in Cahn-Hilliard equations and systems

Abstract

I will discuss the asymptotic behavior of perturbations of transition
front solutions arising in Cahn–Hilliard equations and systems on
$\mathbb{R}$ and $\mathbb{R}^n$.  Such equations arise naturally in
the study of phase separation processes, where a two-phase process
can often be modeled by a Cahn-Hilliard equation, while a process
with more than two phases can be modeled by a Cahn-Hilliard system.
When a Cahn–Hilliard equation or system is linearized about
a transition front solution, the linearized operator has an
eigenvalue at 0 (due to shift invariance), which is not separated
from essential spectrum.  In many cases, it’s possible to
verify that the remaining spectrum lies on the negative
real axis, so that stability is entirely determined by the
nature of this leading eigenvalue.  Working primarily in
the case of a single equation on $\mathbb{R}$, I will discuss
the nature of this leading eigenvalue and also the verification
that spectral stability—defined in terms of an
appropriate Evans function—implies phase-asymptotic stability.
Results for scalar and multidimensional systems will be
summarized.