21,882 research outputs found
Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation
In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combination of the theory introduced in Global smooth solution curves using rigorous branch following (van den Berg et al., Math. Comput. 79(271):1565-1584, 2010) and in Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs (Gameiro and Lessard, J. Diff. Equ. 249(9):2237-2268, 2010). Using this method, one can obtain proofs of existence of global smooth solution curves of equilibria for large (continuous) parameter ranges and about local uniqueness of the solutions on the curve. As an application, we compute several smooth branches of equilibria for the three-dimensional Cahn-Hilliard equation
Mean Field Limits for Interacting Diffusions in a Two-Scale Potential
In this paper we study the combined mean field and homogenization limits for
a system of weakly interacting diffusions moving in a two-scale, locally
periodic confining potential, of the form considered
in~\cite{DuncanPavliotis2016}. We show that, although the mean field and
homogenization limits commute for finite times, they do not, in general,
commute in the long time limit. In particular, the bifurcation diagrams for the
stationary states can be different depending on the order with which we take
the two limits. Furthermore, we construct the bifurcation diagram for the
stationary McKean-Vlasov equation in a two-scale potential, before passing to
the homogenization limit, and we analyze the effect of the multiple local
minima in the confining potential on the number and the stability of stationary
solutions
Hopf bifurcation from fronts in the Cahn-Hilliard equation
We study Hopf bifurcation from traveling-front solutions in the Cahn-Hilliard
equation. The primary front is induced by a moving source term. Models of this
form have been used to study a variety of physical phenomena, including pattern
formation in chemical deposition and precipitation processes. Technically, we
study bifurcation in the presence of essential spectrum. We contribute a simple
and direct functional analytic method and determine bifurcation coefficients
explicitly. Our approach uses exponential weights to recover Fredholm
properties and spectral flow ideas to compute Fredholm indices. Simple mass
conservation helps compensate for negative indices. We also construct an
explicit, prototypical example, prove the existence of a bifurcating front, and
determine the direction of bifurcation
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