2,490 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
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
Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model
We investigate the long time behavior of the critical mass
Patlak-Keller-Segel equation. This equation has a one parameter family of
steady-state solutions , , with thick tails whose
second moment is not bounded. We show that these steady state solutions are
stable, and find basins of attraction for them using an entropy functional
coming from the critical fast diffusion equation in
. We construct solutions of Patlak-Keller-Segel equation satisfying an
entropy-entropy dissipation inequality for . While the
entropy dissipation for is strictly positive, it turns
out to be a difference of two terms, neither of which need to be small when the
dissipation is small. We introduce a strategy of "controlled concentration" to
deal with this issue, and then use the regularity obtained from the
entropy-entropy dissipation inequality to prove the existence of basins of
attraction for each stationary state composed by certain initial data
converging towards . In the present paper, we do not provide any
estimate of the rate of convergence, but we discuss how this would result from
a stability result for a certain sharp Gagliardo-Nirenberg-Sobolev inequality.Comment: This version of the paper improves on the previous version by
removing the small size condition on the value of the second Lyapunov
functional of the initial data. The improved methodology makes greater use of
techniques from optimal mass transportation, and so the second and third
sections have changed places, and the current third section completely
rewritte
On global minimizers of repulsive-attractive power-law interaction energies
We consider the minimisation of power-law repulsive-attractive interaction
energies which occur in many biological and physical situations. We show
existence of global minimizers in the discrete setting and get bounds for their
supports independently of the number of Dirac Deltas in certain range of
exponents. These global discrete minimizers correspond to the stable spatial
profiles of flock patterns in swarming models. Global minimizers of the
continuum problem are obtained by compactness. We also illustrate our results
through numerical simulations.Comment: 14 pages, 2 figure
One dimensional Fokker-Planck reduced dynamics of decision making models in Computational Neuroscience
We study a Fokker-Planck equation modelling the firing rates of two
interacting populations of neurons. This model arises in computational
neuroscience when considering, for example, bistable visual perception problems
and is based on a stochastic Wilson-Cowan system of differential equations. In
a previous work, the slow-fast behavior of the solution of the Fokker-Planck
equation has been highlighted. Our aim is to demonstrate that the complexity of
the model can be drastically reduced using this slow-fast structure. In fact,
we can derive a one-dimensional Fokker-Planck equation that describes the
evolution of the solution along the so-called slow manifold. This permits to
have a direct efficient determination of the equilibrium state and its
effective potential, and thus to investigate its dependencies with respect to
various parameters of the model. It also allows to obtain information about the
time escaping behavior. The results obtained for the reduced 1D equation are
validated with those of the original 2D equation both for equilibrium and
transient behavior
A blob method for diffusion
As a counterpoint to classical stochastic particle methods for diffusion, we
develop a deterministic particle method for linear and nonlinear diffusion. At
first glance, deterministic particle methods are incompatible with diffusive
partial differential equations since initial data given by sums of Dirac masses
would be smoothed instantaneously: particles do not remain particles. Inspired
by classical vortex blob methods, we introduce a nonlocal regularization of our
velocity field that ensures particles do remain particles, and we apply this to
develop a numerical blob method for a range of diffusive partial differential
equations of Wasserstein gradient flow type, including the heat equation, the
porous medium equation, the Fokker-Planck equation, the Keller-Segel equation,
and its variants. Our choice of regularization is guided by the Wasserstein
gradient flow structure, and the corresponding energy has a novel form,
combining aspects of the well-known interaction and potential energies. In the
presence of a confining drift or interaction potential, we prove that
minimizers of the regularized energy exist and, as the regularization is
removed, converge to the minimizers of the unregularized energy. We then
restrict our attention to nonlinear diffusion of porous medium type with at
least quadratic exponent. Under sufficient regularity assumptions, we prove
that gradient flows of the regularized energies converge to solutions of the
porous medium equation. As a corollary, we obtain convergence of our numerical
blob method, again under sufficient regularity assumptions. We conclude by
considering a range of numerical examples to demonstrate our method's rate of
convergence to exact solutions and to illustrate key qualitative properties
preserved by the method, including asymptotic behavior of the Fokker-Planck
equation and critical mass of the two-dimensional Keller-Segel equation
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