7,455 research outputs found
Scalable and Quasi-Contractive Markov Coupling of Maxwell Collision
This paper considers space homogenous Boltzmann kinetic equations in
dimension with Maxwell collisions (and without Grad's cut-off). An explicit
Markov coupling of the associated conservative (Nanbu) stochastic -particle
system is constructed, using plain parallel coupling of isotropic random walks
on the sphere of two-body collisional directions. The resulting coupling is
almost surely decreasing, and the -coupling creation is computed
explicitly. Some quasi-contractive and uniform in coupling / coupling
creation inequalities are then proved, relying on -moments () of velocity distributions; upon -uniform propagation of moments of the
particle system, it yields a -scalable -power law trend to
equilibrium. The latter are based on an original sharp inequality, which bounds
from above the coupling distance of two centered and normalized random
variables in , with the average square parallelogram area spanned
by , denoting an independent copy. Two
counter-examples proving the necessity of the dependance on -moments and
the impossibility of strict contractivity are provided. The paper, (mostly)
self-contained, does not require any propagation of chaos property and uses
only elementary tools.Comment: 29 page
Quasineutral limit for Vlasov-Poisson with Penrose stable data
We study the quasineutral limit of a Vlasov-Poisson system that describes the
dynamics of ions in a plasma. We handle data with Sobolev regularity under the
sharp assumption that the profile of the initial data in the velocity variable
satisfies a Penrose stability condition.
As a by-product of our analysis, we obtain a well-posedness theory for the
limit equation (which is a Vlasov equation with Dirac distribution as
interaction kernel) for such data
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