628 research outputs found
On massless electron limit for a multispecies kinetic system with external magnetic field
We consider a three-dimensional kinetic model for a two species plasma
consisting of electrons and ions confined by an external nonconstant magnetic
field. Then we derive a kinetic-fluid model when the mass ratio tends
to zero. Each species initially obeys a Vlasov-type equation and the
electrostatic coupling follows from a Poisson equation. In our modeling, ions
are assumed non-collisional while a Fokker-Planck collision operator is taken
into account in the electron equation. As the mass ratio tends to zero we show
convergence to a new system where the macroscopic electron density satisfies an
anisotropic drift-diffusion equation. To achieve this task, we overcome some
specific technical issues of our model such as the strong effect of the
magnetic field on electrons and the lack of regularity at the limit. With
methods usually adapted to diffusion limit of collisional kinetic equations and
including renormalized solutions, relative entropy dissipation and velocity
averages, we establish the rigorous derivation of the limit model
Kinetic equations with Maxwell boundary conditions
We prove global stability results of {\sl DiPerna-Lions} renormalized
solutions for the initial boundary value problem associated to some kinetic
equations, from which existence results classically follow. The (possibly
nonlinear) boundary conditions are completely or partially diffuse, which
includes the so-called Maxwell boundary conditions, and we prove that it is
realized (it is not only a boundary inequality condition as it has been
established in previous works). We are able to deal with Boltzmann,
Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace
theorems of the kind previously introduced by the author for the Vlasov
equations, new results concerning weak-weak convergence (the renormalized
convergence and the biting -weak convergence), as well as the
Darroz\`es-Guiraud information in a crucial way
Spatially homogeneous solutions of the Vlasov-Nordstr\"om-Fokker-Planck system
The Vlasov-Nordstr\"{o}m-Fokker-Planck system describes the evolution of
self-gravitating matter experiencing collisions with a fixed background of
particles in the framework of a relativistic scalar theory of gravitation. We
study the spatially-homogeneous system and prove global existence and
uniqueness of solutions for the corresponding initial value problem in three
momentum dimensions. Additionally, we study the long time asymptotic behavior
of the system and prove that even in the absence of friction, solutions possess
a non-trivial asymptotic profile. An exact formula for the long time limit of
the particle density is derived in the ultra-relativistic case.Comment: 25 pages, 1 figure. Several changes from previous version. To appear
in J. Diff. Eq
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