Paola Pietra Eric Polizzi Fabrice Deluzet Jihène Kéfi Olivier Pinaud Claudia Negulescu Nicolas Vauchelet Raymond El Hajj Clément Jourdana Stefan Possanner
Kinetic & Related Models 2011, 4(4): i-iii doi: 10.3934/krm.2011.4.4i
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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Analysis of a diffusive effective mass model for nanowires
Clément Jourdana Nicolas Vauchelet
Kinetic & Related Models 2011, 4(4): 1121-1142 doi: 10.3934/krm.2011.4.1121
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.
keywords: Hamiltonian's spectrum. relative entropy method diffusive limit Drift-diffusion system
Numerical simulation of a kinetic model for chemotaxis
Nicolas Vauchelet
Kinetic & Related Models 2010, 3(3): 501-528 doi: 10.3934/krm.2010.3.501
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.
keywords: Kinetic equations semi-Lagrangian method Chemotaxis convergence analysis.
Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations
François James Nicolas Vauchelet
Discrete & Continuous Dynamical Systems - A 2016, 36(3): 1355-1382 doi: 10.3934/dcds.2016.36.1355
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)) [18], 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)) [26]. However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case.
keywords: gradient flow aggregation equation measure-valued solutions optimal transport. nonlocal conservation equations Duality solutions
Existence and diffusive limit of a two-species kinetic model of chemotaxis
Casimir Emako Luís Neves de Almeida Nicolas Vauchelet
Kinetic & Related Models 2015, 8(2): 359-380 doi: 10.3934/krm.2015.8.359
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.
keywords: Chemotaxis drift-diffusion limit two-species Keller-Segel model. kinetic model
Synchronising and non-synchronising dynamics for a two-species aggregation model
Casimir Emako-Kazianou Jie Liao Nicolas Vauchelet
Discrete & Continuous Dynamical Systems - B 2017, 22(6): 2121-2146 doi: 10.3934/dcdsb.2017088

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.

keywords: Hydrodynamic limit duality solution two-species chemotaxis aggregate dynamics finite volume scheme
Quantifying the survival uncertainty of Wolbachia-infected mosquitoes in a spatial model
Martin Strugarek Nicolas Vauchelet Jorge P. Zubelli
Mathematical Biosciences & Engineering 2018, 15(4): 961-991 doi: 10.3934/mbe.2018043

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.

keywords: Reaction-diffusion equation Wolbachia uncertainty quantification population replacement mosquito release protocol

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