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THESE : Analysis and approximation of BGK models (M/F)

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General information

Reference : UMR5251-MURHAC-003
Workplace : TALENCE
Date of publication : Thursday, May 14, 2020
Scientific Responsible name : Stéphane BRULL
Type of Contract : PhD Student contract / Thesis offer
Contract Period : 36 months
Start date of the thesis : 1 October 2020
Proportion of work : Full time
Remuneration : 2 135,00 € gross monthly

Description of the thesis topic

The complexity of the nonlinear Boltzmann equation needs to introduce simpler models satisfying the main properties (conservation laws, H theorem, equilibrium states, …). The classical BGK model allows for example a full implicitation of the relaxation term. This models corresponds to a relaxation model whose attractors are exponential distributions functions leading to the conservation of mass, momentum and total energy. However, existence theorems are more difficult to obtain due to the exponential nonlinearity of the collision operator.

We aim to obtain existence theorems for a bounded domain in space in a stationary case. In general, the Diperna-Lions theory is only devoted to time dependant problems. Therefore a new theory has been developed for the Boltzmann equation. But this theory is lacking for BGK models. Moreover, some existence results exist for BGK models but only for data close to equilibrium states allowing a perturbative approach.

Next we aim to consider gas mixtures where several BGK models exit. The situation is then much more complicated as in the monospecies case according to the interactions between different species. We aim to consider the BGK model derived by Andries, Aoki and Perthame that is used in numerical simulations. This model satisfies in particular H theorem, nonnegativity of the distribution function. This work will have an implementation part, where the models will be implemented and tested on innovative numerical methods.

Work Context

The Intitute of Mathematics of Bordeaux (IMB) is a mixed unit of research (UMR 5251) associated to CNRS, university of Bordeaux and Bordeaux INP.
The Institute of Mathematics hosts the doctoral school of mathematics and computing science. The IMB gathers most of the research activity of the 136 researchers, of the 102 phd students and post-doctorants, and of the 24 members of the staff devoted to assist research.

Stays in South Korea are to be expected during the contract period.

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