Influence du stochastique sur des problèmatiques de changement d'échelle

The work of this thesis belongs to the field of partial differential equations. More specifically, it is linked to the problematic of scale changes in the context of kinetic of gas. Indeed, knowing that there exists different scales of description for a gas (microscopic, mesoscopic and macroscopic scale), we want to link these different associated scales in a context where some randomness acts, in initial data and/or distributed on all the time interval.
In a first part, we establish the rigorous derivation of the linear Boltzmann equation without cut- off starting from a particle system interacting via a potential of infinite range, under particular assumptions of decay, starting from a perturbed equilibrium.
The second part deals with the passage from a stochastic BGK model with high-field scaling to a scalar conservation law with stochastic forcing. First, we establish the existence of a solution to the considered BGK model. Under an additional assumption, we prove then the convergence to a kinetic formulation associated to the conservation law with stochastic forcing.
In the third part, in a first place, we quantify in the case of discrete velocities the defect of regularity associated to the traditional averaging which are considered in the averaging lemmas. Then, we establish a stochastic averaging lemma in that same case. We apply then the result to the context of Rosseland approximation to establish the diffusive limit associated to this model.
Finally, in the last part, we are interested into the numerical study of Uchiyama’s model of square particles with four velocities in dimension two. After adapting the methods of simulation which were developed in the case of hard spheres, we carry out a statistical study of the limits at different scales of this model. We reject the hypothesis of a fractional Brownian motion as diffusive limit.


Nathalie Ayi
Associate Professor