8,660 research outputs found
Zoology of a non-local cross-diffusion model for two species
We study a non-local two species cross-interaction model with
cross-diffusion. We propose a positivity preserving finite volume scheme based
on the numerical method introduced in Ref. [15] and explore this new model
numerically in terms of its long-time behaviours. Using the so gained insights,
we compute analytical stationary states and travelling pulse solutions for a
particular model in the case of attractive-attractive/attractive-repulsive
cross-interactions. We show that, as the strength of the cross-diffusivity
decreases, there is a transition from adjacent solutions to completely
segregated densities, and we compute the threshold analytically for
attractive-repulsive cross-interactions. Other bifurcating stationary states
with various coexistence components of the support are analysed in the
attractive-attractive case. We find a strong agreement between the numerically
and the analytically computed steady states in these particular cases, whose
main qualitative features are also present for more general potentials
Local well-posedness of the generalized Cucker-Smale model
In this paper, we study the local well-posedness of two types of generalized
Cucker-Smale (in short C-S) flocking models. We consider two different
communication weights, singular and regular ones, with nonlinear coupling
velocities for . For the singular
communication weight, we choose with and in dimension . For the regular case, we
select belonging to (L_{loc}^\infty \cap
\mbox{Lip}_{loc})(\mathbb{R}^d) and . We also
remark the various dynamics of C-S particle system for these communication
weights when
Exponential Convergence Towards Stationary States for the 1D Porous Medium Equation with Fractional Pressure
We analyse the asymptotic behaviour of solutions to the one dimensional
fractional version of the porous medium equation introduced by Caffarelli and
V\'azquez, where the pressure is obtained as a Riesz potential associated to
the density. We take advantage of the displacement convexity of the Riesz
potential in one dimension to show a functional inequality involving the
entropy, entropy dissipation, and the Euclidean transport distance. An argument
by approximation shows that this functional inequality is enough to deduce the
exponential convergence of solutions in self-similar variables to the unique
steady states
The derivation of Swarming models: Mean-Field Limit and Wasserstein distances
These notes are devoted to a summary on the mean-field limit of large
ensembles of interacting particles with applications in swarming models. We
first make a summary of the kinetic models derived as continuum versions of
second order models for swarming. We focus on the question of passing from the
discrete to the continuum model in the Dobrushin framework. We show how to use
related techniques from fluid mechanics equations applied to first order models
for swarming, also called the aggregation equation. We give qualitative bounds
on the approximation of initial data by particles to obtain the mean-field
limit for radial singular (at the origin) potentials up to the Newtonian
singularity. We also show the propagation of chaos for more restricted set of
singular potentials
Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion
Recently, there has been a wide interest in the study of aggregation
equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate
diffusion. The focus of this paper is the unification and generalization of the
well-posedness theory of these models. We prove local well-posedness on bounded
domains for dimensions and in all of space for , the
uniqueness being a result previously not known for PKS with degenerate
diffusion. We generalize the notion of criticality for PKS and show that
subcritical problems are globally well-posed. For a fairly general class of
problems, we prove the existence of a critical mass which sharply divides the
possibility of finite time blow up and global existence. Moreover, we compute
the critical mass for fully general problems and show that solutions with
smaller mass exists globally. For a class of supercritical problems we prove
finite time blow up is possible for initial data of arbitrary mass.Comment: 31 page
Intrinsic noise-induced phase transitions: beyond the noise interpretation
We discuss intrinsic noise effects in stochastic multiplicative-noise partial
differential equations, which are qualitatively independent of the noise
interpretation (Ito vs. Stratonovich), in particular in the context of
noise-induced ordering phase transitions. We study a model which, contrary to
all cases known so far, exhibits such ordering transitions when the noise is
interpreted not only according to Stratonovich, but also to Ito. The main
feature of this model is the absence of a linear instability at the transition
point. The dynamical properties of the resulting noise-induced growth processes
are studied and compared in the two interpretations and with a reference
Ginzburg-Landau type model. A detailed discussion of new numerical algorithms
used in both interpretations is also presented.Comment: 9 pages, 8 figures, to be published in Phys. Rev.
Singular Cucker-Smale Dynamics
The existing state of the art for singular models of flocking is overviewed,
starting from microscopic model of Cucker and Smale with singular communication
weight, through its mesoscopic mean-filed limit, up to the corresponding
macroscopic regime. For the microscopic Cucker-Smale (CS) model, the
collision-avoidance phenomenon is discussed, also in the presence of bonding
forces and the decentralized control. For the kinetic mean-field model, the
existence of global-in-time measure-valued solutions, with a special emphasis
on a weak atomic uniqueness of solutions is sketched. Ultimately, for the
macroscopic singular model, the summary of the existence results for the
Euler-type alignment system is provided, including existence of strong
solutions on one-dimensional torus, and the extension of this result to higher
dimensions upon restriction on the smallness of initial data. Additionally, the
pressureless Navier-Stokes-type system corresponding to particular choice of
alignment kernel is presented, and compared - analytically and numerically - to
the porous medium equation
A Chandra Snapshot Survey of IR-bright LINERs: A Possible Link Between Star Formation, AGN Fueling, and Mass Accretion
We present results from a high resolution X-ray imaging study of nearby
LINERs observed by Chandra. This study complements and extends previous X-ray
studies of LINERs, focusing on the under-explored population of nearby
dust-enshrouded infrared-bright LINERs. The sample consists of 15 IR-bright
LINERs (L_FIR/L_B > 3), with distances that range from 11 to 26 Mpc. Combining
our sample with previous Chandra studies we find that ~ 51% (28/55) of the
LINERs display compact hard X-ray cores. The nuclear 2-10 keV luminosities of
the galaxies in this expanded sample range from ~ 2 X 10^38 ergs s^-1 to ~ 2 X
10^44 ergs s^-1. We find an intriguing trend in the Eddington ratio vs. L_FIR
and L_FIR/L_B for the AGN-LINERs in the expanded sample that extends over seven
orders of magnitude in L/L_Edd. This correlation may imply a link between black
hole growth, as measured by the Eddington ratio, and the star formation rate
(SFR), as measured by the far-IR luminosity and IR-brightness ratio. If the
far-IR luminosity is an indicator of the molecular gas content in our sample of
LINERs, our results may further indicate that the mass accretion rate scales
with the host galaxy's fuel supply. We discuss the potential implications of
our results in the framework of black hole growth and AGN fueling in low
luminosity AGN. (Abridged)Comment: Accepted for publication by ApJ 14 pages, 13 figure
Uncertainty quantification for kinetic models in socio-economic and life sciences
Kinetic equations play a major rule in modeling large systems of interacting
particles. Recently the legacy of classical kinetic theory found novel
applications in socio-economic and life sciences, where processes characterized
by large groups of agents exhibit spontaneous emergence of social structures.
Well-known examples are the formation of clusters in opinion dynamics, the
appearance of inequalities in wealth distributions, flocking and milling
behaviors in swarming models, synchronization phenomena in biological systems
and lane formation in pedestrian traffic. The construction of kinetic models
describing the above processes, however, has to face the difficulty of the lack
of fundamental principles since physical forces are replaced by empirical
social forces. These empirical forces are typically constructed with the aim to
reproduce qualitatively the observed system behaviors, like the emergence of
social structures, and are at best known in terms of statistical information of
the modeling parameters. For this reason the presence of random inputs
characterizing the parameters uncertainty should be considered as an essential
feature in the modeling process. In this survey we introduce several examples
of such kinetic models, that are mathematically described by nonlinear Vlasov
and Fokker--Planck equations, and present different numerical approaches for
uncertainty quantification which preserve the main features of the kinetic
solution.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic
Equations
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