2,713 research outputs found
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
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
A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
We propose a positivity preserving entropy decreasing finite volume scheme
for nonlinear nonlocal equations with a gradient flow structure. These
properties allow for accurate computations of stationary states and long-time
asymptotics demonstrated by suitably chosen test cases in which these features
of the scheme are essential. The proposed scheme is able to cope with
non-smooth stationary states, different time scales including metastability, as
well as concentrations and self-similar behavior induced by singular nonlocal
kernels. We use the scheme to explore properties of these equations beyond
their present theoretical knowledge
Convergence to Equilibrium in Wasserstein distance for damped Euler equations with interaction forces
We develop tools to construct Lyapunov functionals on the space of
probability measures in order to investigate the convergence to global
equilibrium of a damped Euler system under the influence of external and
interaction potential forces with respect to the 2-Wasserstein distance. We
also discuss the overdamped limit to a nonlocal equation used in the modelling
of granular media with respect to the 2-Wasserstein distance, and provide
rigorous proofs for particular examples in one spatial dimension
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