On July 5-th 2010, Naoufel Ben Abdallah tragically passed away at the age of 41.
He was an extremely talented mathematician with a deep interest in applications,
and an uncommon ability in creating links for interdisciplinary research.
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We propose in this paper to derive and analyze a self-consistent model describing the diffusive transport in a nanowire. From a physical point of view, it describes the electron transport in an ultra-scaled confined structure, taking into account the interactions of charged particles with phonons. The transport direction is assumed to be large compared to the wire section and is described by a drift-diffusion equation including effective quantities computed from a Bloch problem in the crystal lattice.
The electrostatic potential solves a Poisson equation where the particle
density couples on each energy band a two dimensional confinement density with the monodimensional transport density given by the Boltzmann statistics.
On the one hand, we study the derivation of this Nanowire Drift-Diffusion Poisson model from a kinetic level description. On the other hand, we present an existence result for this model in a bounded domain.
This paper is devoted to numerical simulations of a kinetic model
describing chemotaxis. This kinetic framework has been investigated
since the 80's when experimental observations have shown that the
motion of bacteria is due to the alternance of 'runs and tumbles'.
Since parabolic and hyperbolic models do not take into account the
microscopic movement of individual cells, kinetic models have become of a
great interest. Dolak and Schmeiser (2005) have then proposed a kinetic
model describing the motion of bacteria responding to temporal gradients
of chemoattractants along their paths.
An existence result for this system is provided and a
numerical scheme relying on a semi-Lagrangian method is presented
and analyzed. An implementation of this scheme allows to obtain
numerical simulations of the model and observe blow-up patterns that
differ greatly from the case of Keller-Segel type of models.
Existence and uniqueness of global in time measure solution for a
one dimensional nonlinear aggregation equation is considered.
Such a system can be written as a conservation law with a
velocity field computed through a self-consistent interaction potential.
Blow up of regular solutions is now well established for such
system. In Carrillo et al. (Duke Math J (2011)) ,
a theory of existence and uniqueness based on the geometric
approach of gradient flows on Wasserstein space has been developed.
We propose in this work to establish the link between this approach
and duality solutions. This latter concept of solutions
allows in particular to define a flow associated to the velocity field.
Then an existence and uniqueness theory for duality solutions is
developed in the spirit of James and Vauchelet (NoDEA (2013)) .
However, since duality solutions are only known in one dimension, we
restrict our study to the one dimensional case.
In this paper, we propose a kinetic model describing the collective motion
by chemotaxis of two species in interaction emitting the same chemoattractant.
Such model can be seen as a generalisation to several species
of the Othmer-Dunbar-Alt model which takes into account the run-and-tumble
process of bacteria.
Existence of weak solutions for this two-species kinetic model is studied and
the convergence of its diffusive limit towards a macroscopic model of
Keller-Segel type is analysed.
This paper deals with analysis and numerical simulations of a one-dimensional two-species hyperbolic aggregation model. This model is formed by a system of transport equations with nonlocal velocities, which describes the aggregate dynamics of a two-species population in interaction appearing for instance in bacterial chemotaxis. Blow-up of classical solutions occurs in finite time. This raises the question to define measure-valued solutions for this system. To this aim, we use the duality method developed for transport equations with discontinuous velocity to prove the existence and uniqueness of measure-valued solutions. The proof relies on a stability result. In addition, this approach allows to study the hyperbolic limit of a kinetic chemotaxis model. Moreover, we propose a finite volume numerical scheme whose convergence towards measure-valued solutions is proved. It allows for numerical simulations capturing the behaviour after blow up. Finally, numerical simulations illustrate the complex dynamics of aggregates until the formation of a single aggregate: after blow-up of classical solutions, aggregates of different species are synchronising or nonsynchronising when collide, that is move together or separately, depending on the parameters of the model and masses of species involved.
Artificial releases of Wolbachia-infected Aedes mosquitoes have been under study in the past yearsfor fighting vector-borne diseases such as dengue, chikungunya and zika.Several strains of this bacterium cause cytoplasmic incompatibility (CI) and can also affect their host's fecundity or lifespan, while highly reducing vector competence for the main arboviruses.
We consider and answer the following questions: 1) what should be the initial condition (i.e. size of the initial mosquito population) to have invasion with one mosquito release source? We note that it is hard to have an invasion in such case. 2) How many release points does one need to have sufficiently high probability of invasion? 3) What happens if one accounts for uncertainty in the release protocol (e.g. unequal spacing among release points)?
We build a framework based on existing reaction-diffusion models for the uncertainty quantification in this context,obtain both theoretical and numerical lower bounds for the probability of release successand give new quantitative results on the one dimensional case.
The nonlocal nonlinear aggregation equation in one space dimension is investigated.
In the so-called attractive case smooth solutions blow up in finite time, so that weak measure solutions are introduced.
The velocity involved in the equation becomes discontinuous, and a particular care has to be paid to its definition
as well as the formulation of the corresponding flux. When this is done, the notion of duality solutions allows to
obtain global in time existence and uniqueness for measure solutions. An upwind finite volume scheme is also analyzed,
and the convergence towards the unique solution is proved. Numerical examples show the
dynamics of the solutions after the blow up time.