6,223 research outputs found
Refined Asymptotics for the subcritical Keller-Segel system and Related Functional Inequalities
We analyze the rate of convergence towards self-similarity for the
subcritical Keller-Segel system in the radially symmetric two-dimensional case
and in the corresponding one-dimensional case for logarithmic interaction. We
measure convergence in Wasserstein distance. The rate of convergence towards
self-similarity does not degenerate as we approach the critical case. As a
byproduct, we obtain a proof of the logarithmic Hardy-Littlewood-Sobolev
inequality in the one dimensional and radially symmetric two dimensional case
based on optimal transport arguments. In addition we prove that the
one-dimensional equation is a contraction with respect to Fourier distance in
the subcritical case
Global Solutions for the One-Dimensional Vlasov-Maxwell System for Laser-Plasma Interaction
We analyse a reduced 1D Vlasov--Maxwell system introduced recently in the
physical literature for studying laser-plasma interaction. This system can be
seen as a standard Vlasov equation in which the field is split in two terms: an
electrostatic field obtained from Poisson's equation and a vector potential
term satisfying a nonlinear wave equation. Both nonlinearities in the Poisson
and wave equations are due to the coupling with the Vlasov equation through the
charge density. We show global existence of weak solutions in the
non-relativistic case, and global existence of characteristic solutions in the
quasi-relativistic case. Moreover, these solutions are uniquely characterised
as fixed points of a certain operator. We also find a global energy functional
for the system allowing us to obtain -nonlinear stability of some
particular equilibria in the periodic setting
Explicit Equilibrium Solutions For the Aggregation Equation with Power-Law Potentials
Despite their wide presence in various models in the study of collective
behaviors, explicit swarming patterns are difficult to obtain. In this paper,
special stationary solutions of the aggregation equation with power-law kernels
are constructed by inverting Fredholm integral operators or by employing
certain integral identities. These solutions are expected to be the global
energy stable equilibria and to characterize the generic behaviors of
stationary solutions for more general interactions
Discussing a teacher MKT and its role on teacher practice when exploring data analysis
This article considers teacher knowledge in managing mathematically critical situations and the role of what can be termed a mathematical summary in the analysis of a teaching episode, viewed from the perspective of Mathematical Knowledge for Teaching (MKT). The analysis is based on an episode of content review, from a perspective which aims to understand the teacher’s logic rather than merely identify gaps in their knowledge. We discuss the importance of approaching mathematically critical situations in order to contribute to eradicating mathematical innumeracy (statistics) and to promote a kind of practice which is “mathematically demanding” as well as “pedagogically exciting”
Mean-field limit for the stochastic Vicsek model
We consider the continuous version of the Vicsek model with noise, proposed
as a model for collective behavior of individuals with a fixed speed. We
rigorously derive the kinetic mean-field partial differential equation
satisfied when the number N of particles tends to infinity, quantifying the
convergence of the law of one particle to the solution of the PDE. For this we
adapt a classical coupling argument to the present case in which both the
particle system and the PDE are defined on a surface rather than on the whole
space. As part of the study we give existence and uniqueness results for both
the particle system and the PDE
Stochastic Mean-Field Limit: Non-Lipschitz Forces \& Swarming
We consider general stochastic systems of interacting particles with noise
which are relevant as models for the collective behavior of animals, and
rigorously prove that in the mean-field limit the system is close to the
solution of a kinetic PDE. Our aim is to include models widely studied in the
literature such as the Cucker-Smale model, adding noise to the behavior of
individuals. The difficulty, as compared to the classical case of globally
Lipschitz potentials, is that in several models the interaction potential
between particles is only locally Lipschitz, the local Lipschitz constant
growing to infinity with the size of the region considered. With this in mind,
we present an extension of the classical theory for globally Lipschitz
interactions, which works for only locally Lipschitz ones
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