Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
type: Item
instance of: scholarly article
title: Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
zbMATH Open document ID: 1235.35034
author: José A. Cañizo
author: Stéphane Mischler
author: Maria José Cáceres
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2011-10-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.5461
review text: It is by now a well-known fact that, under fairly general assumptions on the data, there is \(\lambda>0\) such that the growth-fragmentation equation \[ \partial_t g + \partial_x (a g) + \lambda g = \mathcal{L}[g]\,, \qquad (t,x)\in (0,\infty)\times (0,\infty)\,, \] with boundary condition \(g(t,0)=0\), \(t>0\), has a unique nonnegative and integrable stationary solution \(G\) satisfying \(\|G\|_{L^1}=1\). Here, the growth rate \(a\) is a nonnegative function of \(x\) and the fragmentation operator \(\mathcal{L}\) is given by \[ \mathcal{L}[g](x) = \int_x^\infty b(y,x) g(y)\;dy - B(x) g(x)\,, \quad B(x) =\int_0^x \frac{y}{x} b(x,y)\;dy\,, \quad x>0\,. \] It is then expected that, given a nonnegative and integrable initial condition \(g_{in}\) satisfying \(\|g_{in}\|_{L^1}=1\), the corresponding solution \(g(t)\) to the growth-fragmentation equation converges to \(G\) as \(t\to\infty\). This has been shown to be true in some cases, and the paper under review focuses on establishing a rate of convergence to zero of \((g(t)-G)\). Assuming that \(a(x)=x^\alpha\), \(\alpha\in\{0,1\}\), \(\int_0^x b(x,y)\;dy = \kappa B(x)\) for some \(\kappa>1\), and \(2B_m x^{\gamma-1} \leq b(x,y) \leq 2B_M x^{\gamma-1}\), \(0<y<x\), for some \(\gamma>-\alpha\) and \(B_M\geq B_m>0\), it is proved that \((g(t)-G)\) decays exponentially fast to zero as \(t\to 0\) in a suitable topology provided \(\gamma\in (0,2)\) or \((\alpha,\gamma)=(1,2)\) or \((\alpha,\gamma)=(0,0)\) and \(B_m=B_M\). The cornerstone of the proof is to establish a so-called entropy/entropy dissipation inequality of the form \(H_2[g|G] \leq C\;D^b[g|G]\) for some \(C>0\), where \(H_2[g|G]=\int_0^\infty (g-G)^2 (\phi/G)\;dx\) is a Liapunov functional for the growth-fragmentation equation for a suitably chosen weight function \(\phi\) and \(D^b[g|G]\) is the associated dissipation (that is, \(dH_2[g|G]/dt = -D^b[g|G]\)).
reviewed by: Philippe Laurençot
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 35F16
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 5956998
zbMATH Keywords: fragmentation
zbMATH Keywords: growth
zbMATH Keywords: large time behaviour
zbMATH Keywords: exponential decay
zbMATH Keywords: functional inequality
zbMATH Keywords: entropy/entropy dissipation inequality
MaRDI profile type: Publication
OpenAlex ID: W1994983983
arXiv ID: 1010.5461
cites work: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials
cites work: Geometric bounds for eigenvalues of Markov chains
cites work: EIGENELEMENTS OF A GENERAL AGGREGATION-FRAGMENTATION MODEL
cites work: On self-similarity and stationary problem for fragmentation and coagulation models.
cites work: Exponential decay for the growth-fragmentation/cell-division equations
cites work: Q4723599
cites work: EXISTENCE OF A SOLUTION TO THE CELL DIVISION EIGENPROBLEM
cites work: General entropy equations for structured population models and scattering
cites work: General relative entropy inequality: an illustration on growth models
cites work: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials
cites work: Q3412821
cites work: Exponential decay for the fragmentation or cell-division equation
DOI: 10.1016/J.MATPUR.2011.01.003
description: scientific article