STABILITY OF THE MAXWELL-STEFAN SYSTEM IN THE DIFFUSION ASYMPTOTICS OF THE BOLTZMANN MULTI-SPECIES EQUATION

We investigate the diffusion asymptotics of the Boltzmann equation for gaseous mixtures, in the perturbative regime around a local Maxwellian vector whose fluid quantities solve a flux-incompressible Maxwell-Stefan system. Our framework is the torus and we consider hard-potential collision kernels with angular cutoff. As opposed to existing results about hydrodynamic limits in the mono-species case, the local Maxwellian we study here is not a local equilibrium of the mixture due to cross-interactions. By means of a hypocoercive formalism and introducing a suitable modified Sobolev norm, we build a Cauchy theory which is uniform with respect to the Knudsen number ε. In this way, we shall prove that the Maxwell-Stefan system is stable for the Boltzmann multi-species equation, ensuring a rigorous derivation in the vanishing limit ε → 0

ISOTROPIC DIFFEOMORPHISMS: SOLUTIONS TO A DIFFERENTIAL SYSTEM FOR A DEFORMED RANDOM FIELDS STUDY

This Note presents the resolution of a differential system on the plane that translates a geometrical problem about isotropic deformations of area and length. The system stems from a probability study on deformed random fields [1], which are the composition of a random field with invariance properties defined on the plane with a deterministic diffeomorphism. The explicit resolution of the differential system allows to prove that a weak notion of isotropy of the deformed field, linked to its excursion sets, in fact coincides with the strong notion of isotropy. The present Note first introduces the probability framework that gave rise to the geometrical issue and then proposes its resolution

PERTURBATIVE CAUCHY THEORY FOR A FLUX-INCOMPRESSIBLE MAXWELL-STEFAN SYSTEM IN A NON-EQUIMOLAR REGIME

We establish a quantitative Cauchy theory in Sobolev spaces for the Maxwell-Stefan equations with an incompressibility condition on the total flux. More precisely, we prove existence, uniqueness in a weak sense and exponential trend to equilibrium of solutions in a perturbative regime around any macroscopic equilibrium state of the mixture, not necessarily constant. In particular, an orthogonal viewpoint that we found specific to the incompressible setting, combined with the use of a suitable anisotropic norm, allows us to get rid of the usual closure assumption of equimolar diffusion

On the Boltzmann equation, quantitative studies and hydrodynamical limits

A revised version of this thesis, correcting typing errors in the pages indexes and the bibliography, replaced the original version on 17 June 2015The present thesis deals with the mathematical treatment of kinetic theory and focuses more precisely on the Boltzmann equation. We investigate several properties of the solutions to the latter equation: their positivity and their hydrodynamical limits for instance. We also study the local Cauchy problem for a quantic version of the Boltzmann equation.This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis