501 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
Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion
We present a new a-priori estimate for discrete coagulation-fragmentation
systems with size-dependent diffusion within a bounded, regular domain confined
by homogeneous Neumann boundary conditions. Following from a duality argument,
this a-priori estimate provides a global bound on the mass density and
was previously used, for instance, in the context of reaction-diffusion
equations.
In this paper we demonstrate two lines of applications for such an estimate:
On the one hand, it enables to simplify parts of the known existence theory and
allows to show existence of solutions for generalised models involving
collision-induced, quadratic fragmentation terms for which the previous
existence theory seems difficult to apply. On the other hand and most
prominently, it proves mass conservation (and thus the absence of gelation) for
almost all the coagulation coefficients for which mass conservation is known to
hold true in the space homogeneous case.Comment: 24 page
Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations
We establish a weak-strong uniqueness principle for solutions to
entropy-dissipating reaction-diffusion equations: As long as a strong solution
to the reaction-diffusion equation exists, any weak solution and even any
renormalized solution must coincide with this strong solution. Our assumptions
on the reaction rates are just the entropy condition and local Lipschitz
continuity; in particular, we do not impose any growth restrictions on the
reaction rates. Therefore, our result applies to any single reversible reaction
with mass-action kinetics as well as to systems of reversible reactions with
mass-action kinetics satisfying the detailed balance condition. Renormalized
solutions are known to exist globally in time for reaction-diffusion equations
with entropy-dissipating reaction rates; in contrast, the global-in-time
existence of weak solutions is in general still an open problem - even for
smooth data - , thereby motivating the study of renormalized solutions. The key
ingredient of our result is a careful adjustment of the usual relative entropy
functional, whose evolution cannot be controlled properly for weak solutions or
renormalized solutions.Comment: 32 page
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
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
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