The Spitzer-Härm regime arising in plasma physics leads asymptotically to a nonlinear diffusion equation for the electron temperature.
In this work we propose a hierarchy of models intended to retain more features of the underlying modeling based on kinetic equations.
These models are of non--local type. Nevertheless, owing to energy discretization they can lead to coupled systems of diffusion equations.
We make the connection between the different models precise and bring out some mathematical properties of the models.
A numerical scheme is designed for the approximate models, and simulations validate the proposed approach.
We are interested in non-standard transport equations where the description of the scattering events involves an additional "memory variable''.
We establish the well posedness and investigate the diffusion asymptotics of such models.
While the questions we address are quite classical the analysis is original
since the usual dissipative properties of collisional transport equations is broken
by the introduction of the memory terms.
We study the asymptotic regime for the relativistic
Vlasov-Maxwell-Fokker-Planck system which corresponds to a mean
free path small compared to the Debye length, chosen as an
observation length scale, combined to a large thermal velocity
assumption. We are led to a convection-diffusion equation, where
the convection velocity is obtained by solving a Poisson equation.
The analysis is performed in the one and one half dimensional case
and the proof combines dissipation mechanisms and finite speed of
We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicle density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions.
Cancer is one of the greatest killers in
the world, particularly in western countries. A lot of the effort of
the medical research is devoted to cancer and mathematical modeling
must be considered as an additional tool for the physicians and
biologists to understand cancer mechanisms and to determine the
adapted treatments. Metastases make all the seriousness of cancer. In
2000, Iwata et al.  proposed a model which describes the
evolution of an untreated metastatic tumors population. We provide
here a mathematical analysis of this model which brings us to the
determination of a Malthusian rate characterizing the exponential
growth of the population. We provide as well a numerical analysis of the PDE given by the model.
The Lifschitz--Slyozov system describes the dynamics of mass exchanges between
macro--particles and monomers in the theory of coarsening.
We consider a variant of the classical model where monomers are subject to space diffusion.
We establish the existence--uniqueness of solutions for a wide class of relevant data and kinetic coefficients.
We also derive a numerical scheme to simulate the behavior of the solutions.