Analysis of a chemotaxis system modeling ant foraging

In this paper we analyze a system of PDEs recently introduced in [P. Amorim, {\it Modeling ant foraging: a {chemotaxis} approach with pheromones and trail formation}], in order to describe the dynamics of ant foraging. The system is made of convection-diffusion-reaction equations, and the coupling is driven by chemotaxis mechanisms. We establish the well-posedness for the model, and investigate the regularity issue for a large class of integrable data. Our main focus is on the (physically relevant) two-dimensional case with boundary conditions, where we prove that the solutions remain bounded for all times. The proof involves a series of fine \emph{a priori} estimates in Lebesgue spaces.Comment: 39 page

Low Field Regime for the Relativistic Vlasov-Maxwell-Fokker-Planck System; the One and One Half Dimensional Case

International audienceWe study the asymptotic regime for the relativistic Vlasov-Maxwell-Fokker-Planck system which corresponds to a mean free path small compared to the Debye length, chosen as an observation length scale, combined to a large thermal velocity assumption. We are led to a convection-diffusion equation, where the convection velocity is obtained by solving a Poisson equation. The analysis is performed in the one and one half dimensional case and the proof combines dissipation mechanisms and finite speed of propagation properties

On a Fokker-Planck equation arising in population dynamics

Sin resume

From Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck Systems to Incompressible Euler Equations: the case with finite charge

We study the asymptotic regime of strong electric fields that leads from the Vlasov-Poisson system to the Incompressible Euler equations. We also deal with the Vlasov-Poisson-Fokker-Planck system which induces dissipative effects. The originality consists in considering a situation with a finite total charge confined by a strong external field. In turn, the limiting equation is set in a bounded domain, the shape of which is determined by the external confining potential. The analysis extends to the situation where the limiting density is non-homogeneous and where the Euler equation is replaced by the Lake Equation, also called Anelastic Equation.Comment: 39 pages, 3 figure

A Numerical Study on Large-Time Asymptotics of the Lifshitz-Slyozov System

We numerically investigate the behaviour for long time of solutions of the Lifshitz-Slyozov system. In particular, we find this behaviour to crucially depend on the distribution of largest aggregates present in the solution

Discrete Version of the She Asymptotics: Multigroup Neutron Transport Equations

This paper is devoted to the derivation of multigroup diffusion equations from the Boltzmann equation. The limit system couples the energy levels from both zeroth order term and diffusion currents

Plane wave stability analysis of Hartree and quantum dissipative systems

We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment. The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential; such a coupling lead to challenging issues compared to the usual non linear Schr{\"o}dinger equations. The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals. We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime. In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter

Vanishing pressure in gas dynamics equations

International audienceThe smooth solutions of gas dynamics system as the pressure goes to 0 converge toward a solution of the pressureless gas model