165 research outputs found
Existence of Compactly Supported Global Minimisers for the Interaction Energy
The existence of compactly supported global minimisers for continuum models
of particles interacting through a potential is shown under almost optimal
hypotheses. The main assumption on the potential is that it is catastrophic, or
not H-stable, which is the complementary assumption to that in classical
results on thermodynamic limits in statistical mechanics. The proof is based on
a uniform control on the local mass around each point of the support of a
global minimiser, together with an estimate on the size of the "gaps" it may
have. The class of potentials for which we prove existence of global minimisers
includes power-law potentials and, for some range of parameters, Morse
potentials, widely used in applications. We also show that the support of local
minimisers is compact under suitable assumptions.Comment: Final version after referee reports taken into accoun
Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
AbstractWe study the asymptotic behavior of linear evolution equations of the type ∂tg=Dg+Lg−λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg+Lg. In the case Dg=−∂xg, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=−∂x(xg), it is known that λ=1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂tf=Lf.By means of entropy–entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part
Rate of convergence to self-similarity for the fragmentation equation in L^1 spaces
In a recent result by the authors (ref. [1]) it was proved that solutions of
the self-similar fragmentation equation converge to equilibrium exponentially
fast. This was done by showing a spectral gap in weighted spaces of the
operator defining the time evolution. In the present work we prove that there
is also a spectral gap in weighted spaces, thus extending exponential
convergence to a larger set of initial conditions. The main tool is an
extension result in ref. [4]
Existence and approximation of probability measure solutions to models of collective behaviors
In this paper we consider first order differential models of collective
behaviors of groups of agents based on the mass conservation equation. Models
are formulated taking the spatial distribution of the agents as the main
unknown, expressed in terms of a probability measure evolving in time. We
develop an existence and approximation theory of the solutions to such models
and we show that some recently proposed models of crowd and swarm dynamics fit
our theoretic paradigm.Comment: 31 pages, 1 figur
Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance
Under the condition of detailed balance and some additional restrictions on
the size of the coefficients, we identify the equilibrium distribution to which
solutions of the discrete coagulation-fragmentation system of equations
converge for large times, thus showing that there is a critical mass which
marks a change in the behavior of the solutions. This was previously known only
for particular cases as the generalized Becker-D\"oring equations. Our proof is
based on an inequality between the entropy and the entropy production which
also gives some information on the rate of convergence to equilibrium for
solutions under the critical mass.Comment: 28 page
Three eras of micellization
Micellization is the precipitation of lipids from aqueous solution into aggregates with a broad distribution of aggregation number. Three eras of micellization are characterized in a simple kinetic model of Becker-Döring type. The model asigns the same constant energy to the (k-1) monomer-monomer bonds in a linear chain of k particles. The number of monomers decreases sharply and many clusters of small size are produced during the first era. During the second era, nuclei are increasing steadily in size until their distribution becomes a self-similar solution of the diffusion equation. Lastly, when the average size of the nuclei becomes comparable to its equilibrium value, a simple mean-field Fokker-Planck equation describes the final era until the equilibrium distribution is reached
A well-posedness theory in measures for some kinetic models of collective motion
We present existence, uniqueness and continuous dependence results for some
kinetic equations motivated by models for the collective behavior of large
groups of individuals. Models of this kind have been recently proposed to study
the behavior of large groups of animals, such as flocks of birds, swarms, or
schools of fish. Our aim is to give a well-posedness theory for general models
which possibly include a variety of effects: an interaction through a
potential, such as a short-range repulsion and long-range attraction; a
velocity-averaging effect where individuals try to adapt their own velocity to
that of other individuals in their surroundings; and self-propulsion effects,
which take into account effects on one individual that are independent of the
others. We develop our theory in a space of measures, using mass transportation
distances. As consequences of our theory we show also the convergence of
particle systems to their corresponding kinetic equations, and the
local-in-time convergence to the hydrodynamic limit for one of the models
Iron Status Biomarkers and Cardiovascular Risk
Both iron excess and deficiency may be related to oxidative stress. Serum ferritin, the main marker of iron status, and hepcidin, the key regulator of iron metabolism, are increased in inflammation states and their links with insulin resistance are emerging topics. We have reviewed the role of iron deficiency/overload in cardiovascular risk, including our own results. Most studies deal with the association between iron deposition in tissues and cardiovascular risk, while decreased iron status is predominantly related to protection against atherosclerosis and coronary heart disease. Less information is available on the role of iron status in type 2 diabetes mellitus (T2DM). Serum ferritin is positively correlated with several indicators of cardiovascular risk in healthy adults and diabetics, thus excess body iron is related to cardiometabolic alterations including vascular and heart damage, central obesity, and metabolic syndrome. Our data in an ample sample of T2DM adults suggest that body iron stores, evaluated as ferritin, are clearly related with some key markers of the so-called lipidic triad (high triglyceride and low high-density lipoprotein (HDL) cholesterol) levels together with the presence of small and dense low-density lipoprotein particles which also is in the frame of the dysmetabolic iron overload syndrome
One-dimensional inelastic Boltzmann equation: Regularity \& uniqueness of self-similar profiles for moderately hard potentials
We prove uniqueness of self-similar profiles for the one-dimensional
inelastic Boltzmann equation with moderately hard potentials, that is with
collision kernel of the form | | for > 0 small
enough (explicitly quantified). Our result provides the first uniqueness
statement for self-similar profiles of inelastic Boltzmann models allowing for
strong inelasticity besides the explicitly solvable case of Maxwell
interactions (corresponding to = 0). Our approach relies on a
perturbation argument from the corresponding Maxwell model through a careful
study of the associated linearised operator. In particular, a part of the paper
is devoted to the trend to equilibrium for the Maxwell model in suitable
weighted Sobolev spaces, an extension of results which are known to hold in
weaker topologies. Our results can be seen as a first step towards a full
proof, in the one-dimensional setting, of a conjecture in Ernst \& Brito (2002)
regarding the determination of the long-time behaviour of solutions to
inelastic Boltzmann equation
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