107 research outputs found
Homogenization of the discrete diffusive coagulation-fragmentation equations in perforated domains
The asymptotic behavior of the solution of an infinite set of Smoluchowski's
discrete coagulation-fragmentation-diffusion equations with non-homogeneous
Neumann boundary conditions, defined in a periodically perforated domain, is
analyzed. Our homogenization result, based on Nguetseng-Allaire two-scale
convergence, is meant to pass from a microscopic model (where the physical
processes are properly described) to a macroscopic one (which takes into
account only the effective or averaged properties of the system). When the
characteristic size of the perforations vanishes, the information given on the
microscale by the non-homogeneous Neumann boundary condition is transferred
into a global source term appearing in the limiting (homogenized) equations.
Furthermore, on the macroscale, the geometric structure of the perforated
domain induces a correction in the diffusion coefficients
Estimates for the large time behavior of the Landau equation in the Coulomb case
This work deals with the large time behaviour of the spatially homogeneous
Landau equation with Coulomb potential. Firstly, we obtain a bound from below
of the entropy dissipation by a weighted relative Fisher information of
with respect to the associated Maxwellian distribution, which leads to a
variant of Cercignani's conjecture thanks to a logarithmic Sobolev inequality.
Secondly, we prove the propagation of polynomial and stretched exponential
moments with an at most linearly growing in time rate. As an application of
these estimates, we show the convergence of any (- or weak) solution to the
Landau equation with Coulomb potential to the associated Maxwellian equilibrium
with an explicitly computable rate, assuming initial data with finite mass,
energy, entropy and some higher -moment. More precisely, if the initial
data have some (large enough) polynomial -moment, then we obtain an
algebraic decay. If the initial data have a stretched exponential -moment,
then we recover a stretched exponential decay
Entropy, Duality and Cross Diffusion
This paper is devoted to the use of the entropy and duality methods for the
existence theory of reaction-cross diffusion systems consisting of two
equations, in any dimension of space. Those systems appear in population
dynamics when the diffusion rates of individuals of two species depend on the
concentration of individuals of the same species (self-diffusion), or of the
other species (cross diffusion)
Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge
In this paper, we extend to the case of initial data constituted of a Dirac
mass plus a bounded density (with finite moments) the theory of Lions and
Perthame [6] for the Vlasov-Poisson equation. Our techniques also provide
polynomially growing in time estimates for moments of the bounded density.Comment: 27 pages; new version: few typos have been corrected, the
introduction has been modifie
Rigorous numerics for nonlinear operators with tridiagonal dominant linear part
We present a method designed for computing solutions of infinite dimensional
non linear operators with a tridiagonal dominant linear part. We
recast the operator equation into an equivalent Newton-like equation , where is an approximate inverse of the derivative
at an approximate solution . We present rigorous
computer-assisted calculations showing that is a contraction near
, thus yielding the existence of a solution. Since does not have an asymptotically diagonal dominant structure, the
computation of is not straightforward. This paper provides ideas for
computing , and proposes a new rigorous method for proving existence of
solutions of nonlinear operators with tridiagonal dominant linear part.Comment: 27 pages, 3 figures, to be published in DCDS-A (Vol. 35, No. 10)
October 2015 issu
Improved duality estimates and applications to reaction-diffusion equations
We present a refined duality estimate for parabolic equations. This estimate
entails new results for systems of reaction-diffusion equations, including
smoothness and exponential convergence towards equilibrium for equations with
quadratic right-hand sides in two dimensions. For general systems in any space
dimension, we obtain smooth solutions of reaction-diffusion systems coming out
of reversible chemistry under an assumption that the diffusion coefficients are
sufficiently close one to another
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