15,004 research outputs found
Motion of a droplet for the mass-conserving stochastic Allen-Cahn equation
We study the stochastic mass-conserving Allen-Cahn equation posed on a
bounded two-dimensional domain with additive spatially smooth space-time noise.
This equation associated with a small positive parameter describes the
stochastic motion of a small almost semicircular droplet attached to domain's
boundary and moving towards a point of locally maximum curvature. We apply
It\^o calculus to derive the stochastic dynamics of the droplet by utilizing
the approximately invariant manifold introduced by Alikakos, Chen and Fusco for
the deterministic problem. In the stochastic case depending on the scaling, the
motion is driven by the change in the curvature of the boundary and the
stochastic forcing. Moreover, under the assumption of a sufficiently small
noise strength, we establish stochastic stability of a neighborhood of the
manifold of droplets in and , which means that with overwhelming
probability the solution stays close to the manifold for very long time-scales
Emergence of steady and oscillatory localized structures in a phytoplankton-nutrient model
Co-limitation of marine phytoplankton growth by light and nutrient, both of
which are essential for phytoplankton, leads to complex dynamic behavior and a
wide array of coherent patterns. The building blocks of this array can be
considered to be deep chlorophyll maxima, or DCMs, which are structures
localized in a finite depth interior to the water column. From an ecological
point of view, DCMs are evocative of a balance between the inflow of light from
the water surface and of nutrients from the sediment. From a (linear)
bifurcational point of view, they appear through a transcritical bifurcation in
which the trivial, no-plankton steady state is destabilized. This article is
devoted to the analytic investigation of the weakly nonlinear dynamics of these
DCM patterns, and it has two overarching themes. The first of these concerns
the fate of the destabilizing stationary DCM mode beyond the center manifold
regime. Exploiting the natural singularly perturbed nature of the model, we
derive an explicit reduced model of asymptotically high dimension which fully
captures these dynamics. Our subsequent and fully detailed study of this model
- which involves a subtle asymptotic analysis necessarily transgressing the
boundaries of a local center manifold reduction - establishes that a stable DCM
pattern indeed appears from a transcritical bifurcation. However, we also
deduce that asymptotically close to the original destabilization, the DCM
looses its stability in a secondary bifurcation of Hopf type. This is in
agreement with indications from numerical simulations available in the
literature. Employing the same methods, we also identify a much larger DCM
pattern. The development of the method underpinning this work - which, we
expect, shall prove useful for a larger class of models - forms the second
theme of this article
Longtime behavior of nonlocal Cahn-Hilliard equations
Here we consider the nonlocal Cahn-Hilliard equation with constant mobility
in a bounded domain. We prove that the associated dynamical system has an
exponential attractor, provided that the potential is regular. In order to do
that a crucial step is showing the eventual boundedness of the order parameter
uniformly with respect to the initial datum. This is obtained through an
Alikakos-Moser type argument. We establish a similar result for the viscous
nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In
this case the validity of the so-called separation property is crucial. We also
discuss the convergence of a solution to a single stationary state. The
separation property in the nonviscous case is known to hold when the mobility
degenerates at the pure phases in a proper way and the potential is of
logarithmic type. Thus, the existence of an exponential attractor can be proven
in this case as well
Stable transformation of tobacco by electroporation: evidence for plasmid concatenation.
Finite to infinite steady state solutions, bifurcations of an integro-differential equation
We consider a bistable integral equation which governs the stationary
solutions of a convolution model of solid--solid phase transitions on a circle.
We study the bifurcations of the set of the stationary solutions as the
diffusion coefficient is varied to examine the transition from an infinite
number of steady states to three for the continuum limit of the
semi--discretised system. We show how the symmetry of the problem is
responsible for the generation and stabilisation of equilibria and comment on
the puzzling connection between continuity and stability that exists in this
problem
On the spectral properties of L_{+-} in three dimensions
This paper is part of the radial asymptotic stability analysis of the ground
state soliton for either the cubic nonlinear Schrodinger or Klein-Gordon
equations in three dimensions. We demonstrate by a rigorous method that the
linearized scalar operators which arise in this setting, traditionally denoted
by L_{+-}, satisfy the gap property, at least over the radial functions. This
means that the interval (0,1] does not contain any eigenvalues of L_{+-} and
that the threshold 1 is neither an eigenvalue nor a resonance. The gap property
is required in order to prove scattering to the ground states for solutions
starting on the center-stable manifold associated with these states. This paper
therefore provides the final installment in the proof of this scattering
property for the cubic Klein-Gordon and Schrodinger equations in the radial
case, see the recent theory of Nakanishi and the third author, as well as the
earlier work of the third author and Beceanu on NLS. The method developed here
is quite general, and applicable to other spectral problems which arise in the
theory of nonlinear equations
Bulk phase behaviour of binary hard platelet mixtures from density functional theory
We investigate isotropic-isotropic, isotropic-nematic and nematic-nematic
phase coexistence in binary mixtures of circular platelets with vanishing
thickness, continuous rotational degrees of freedom and radial size ratios
up to 5. A fundamental measure density functional theory, previously
used for the one-component model, is proposed and results are compared against
those from Onsager theory as a benchmark. For the system
displays isotropic-nematic phase coexistence with a widening of the biphasic
region for increasing values of . For size ratios , we
find demixing into two nematic states becomes stable and an
isotropic-nematic-nematic triple point can occur. Fundamental measure theory
gives a smaller isotropic-nematic biphasic region than Onsager theory and
locates the transition at lower densities. Furthermore, nematic-nematic
demixing occurs over a larger range of compositions at a given value of
than found in Onsager theory. Both theories predict the same
topologies of the phase diagrams. The partial nematic order parameters vary
strongly with composition and indicate that the larger particles are more
strongly ordered than the smaller particles
Induction of tyrosinase by B-lactose in Neurospora crassa
Induction of tyrosinase by lactose in Neurospora crass
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