I'm interested in the theoretical and numerical study of
kinetic equations (Boltzmann, Vlasov, Fokker-Planck...) and in their
applications. These models usually appear when considering plasma
physics, astrophysics, sprays, and more recently are applied, for example,
in bio-mathematics, economie, population dynamics.
This topic concerns the degassing modeling during volcanic eruptions, in particular we study the gas bubbles growth inside the volcano cone. It is nowadays known that the way in which the bubbles growth and form chains inside the magma influences its porosity, and so its permeability, thus favoring explosive or effusive eruptions.
A first study of this geophysical problem is done considering the evolution of a single spherical gas bubble contained in a spherical volume of magma (the influence region). The bubble is characterized by its volume and mass which time evolution depends on the gas flux diffusing from the magma into the bubble. Together with the analysis of the mathematical model we developed a numerical code (MonoDeBuG) approaching the considered system of equation. This mono-disperse model gives very goods agreements for the evolution of a set of bubbles which not merging when compared with experimental data. But when coalescence takes place it is necessary to consider poly-disperse models. We thus propose a kinetic model describing the evolution of a population of bubbles growing by decompression, exsolution and coalescence. The numerical code developed for this new model (PolyDeBuG) gives good results and permits to check different coalescence rates.
This research begun in the framework of the young-researchers ERC project DEMONS leaded by A. Burgisser (ISTERRE, Chambéry).
In the decision-making for bi-stability visual problems, the firing rates evolutions of two populations of interacting neurons can be modeled by means of a system of stochastic differential equations. It is then possible to describe the evolution of the probability distribution function by the associated two-dimensional Fokker-Planck equation. This equation is characterized by a drift term which is not the gradient of a potential function. Hence we cannot give the explicit form of the steady state solution of the corresponding stationary equation. Still, under the hypothesis that the drift is incoming in the domain, we prove the existence, uniqueness and positivity of the steady state and of the solution to the Fokker-Planck equation, as well as its convergence to the steady state, and its exponential rate of convergence.
The macroscopic quantities we are interested with (from a biological point of view) are the reaction times and the performance, and for both we need to well approximate the steady state. Since computational cost are high, we propose a complexity reduction of the model based on its slow-fast characterization. This reduction permits to write a one-dimensional Fokker-Planck equation living on the equilibrium manifold, and its numerical simulations gives results in agreement with experimental behavior.
This research has been developed in the context of the MANDy ANR project, leaded by M. Thieulen (Paris VI) and was done in collaboration with J.A. Carrillo (Imperial College of London), S. Cordier (Orleans) and G. Deco (Barcelona). It has also been the occasion of the co-direction of the thesis by D. Landon.
Recent development on this subject, in collaboration with G. Barles (Tours) and B. Perthame (Paris VI), has carried me to study and discretize Hamilton-Jacobi equations.
Modeling Cadherines Adhesion (PEPS-MBI : MAC)
The cellular adhesion process play a crucial role for the construction of the embryonic and tissular architecture as well as for some cancers. The mathematical modeling of these phenomena yields to both analytical and numerical interesting questions. Cells adhesion is mainly governed by the growth of trans- and cis-contacts between cadherines. Cadherine is a protein which tends to merge on the border of the cells and its aggregate are linked to the actin filaments. The project PEPS-MBI MAC, bringing together mathematicians and biologists, aims to model, study and compare numerical and experimental results dealing with the first contact formations. Our goal is to give a mechanical characterization of this process, so to help biologists to go further in his comprehension and to consider new experimental protocols.
We choose to model this problem by a statistical description of the contacts growth by engagement of cadherines. The unknown functions represents the freely moving and fixed cadherines and satisfy a degenerate system of reaction-diffusion equation, i.e. one of the diffusion coefficient is null. It appears that in order to recover the same kind of aggregate as in experiences, the reaction term must be defined with respect to the distance to the position in which a link is formed. From a mathematical point of you, we define it by means of a convolution product. It is then possible to show that the system of equations should have a non-homogeneous steady state, yielding to the formation of patterns.