77 research outputs found
Exponential convergence to equilibrium for subcritical solutions of the Becker-D\"oring equations
We prove that any subcritical solution to the Becker-D\"{o}ring equations
converges exponentially fast to the unique steady state with same mass. Our
convergence result is quantitative and we show that the rate of exponential
decay is governed by the spectral gap for the linearized equation, for which
several bounds are provided. This improves the known convergence result by
Jabin & Niethammer (see ref. [14]). Our approach is based on a careful spectral
analysis of the linearized Becker-D\"oring equation (which is new to our
knowledge) in both a Hilbert setting and in certain weighted spaces.
This spectral analysis is then combined with uniform exponential moment bounds
of solutions in order to obtain a convergence result for the nonlinear
equation
Mean-field limit for the stochastic Vicsek model
We consider the continuous version of the Vicsek model with noise, proposed
as a model for collective behavior of individuals with a fixed speed. We
rigorously derive the kinetic mean-field partial differential equation
satisfied when the number N of particles tends to infinity, quantifying the
convergence of the law of one particle to the solution of the PDE. For this we
adapt a classical coupling argument to the present case in which both the
particle system and the PDE are defined on a surface rather than on the whole
space. As part of the study we give existence and uniqueness results for both
the particle system and the PDE
Entropy dissipation estimates for the linear Boltzmann operator
We prove a linear inequality between the entropy and entropy dissipation
functionals for the linear Boltzmann operator (with a Maxwellian equilibrium
background). This provides a positive answer to the analogue of Cercignani's
conjecture for this linear collision operator. Our result covers the physically
relevant case of hard-spheres interactions as well as Maxwellian kernels, both
with and without a cut-off assumption. For Maxwellian kernels, the proof of the
inequality is surprisingly simple and relies on a general estimate of the
entropy of the gain operator due to Matthes and Toscani (2012) and Villani
(1998). For more general kernels, the proof relies on a comparison principle.
Finally, we also show that in the grazing collision limit our results allow to
recover known logarithmic Sobolev inequalities
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
A new approach to the creation and propagation of exponential moments in the Boltzmann equation
We study the creation and propagation of exponential moments of solutions to
the spatially homogeneous -dimensional Boltzmann equation. In particular,
when the collision kernel is of the form for
with and
, and assuming the classical cut-off condition integrable in , we prove that there exists
such that moments with weight are finite
for , where only depends on the collision kernel and the initial mass
and energy. We propose a novel method of proof based on a single differential
inequality for the exponential moment with time-dependent coefficients.Comment: 14 pages. Many typos corrected in this revised versio
Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
AbstractWe study the asymptotic behavior of linear evolution equations of the type ∂tg=Dg+Lg−λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg+Lg. In the case Dg=−∂xg, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=−∂x(xg), it is known that λ=1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂tf=Lf.By means of entropy–entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part
Rate of convergence to self-similarity for the fragmentation equation in L^1 spaces
In a recent result by the authors (ref. [1]) it was proved that solutions of
the self-similar fragmentation equation converge to equilibrium exponentially
fast. This was done by showing a spectral gap in weighted spaces of the
operator defining the time evolution. In the present work we prove that there
is also a spectral gap in weighted spaces, thus extending exponential
convergence to a larger set of initial conditions. The main tool is an
extension result in ref. [4]
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