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Exponential Decay to Equilibrium for a Fiber Lay-Down Process on a Moving Conveyor Belt

Bouin, Emeric and Hoffmann, Franca and Mouhot, Clément (2017) Exponential Decay to Equilibrium for a Fiber Lay-Down Process on a Moving Conveyor Belt. SIAM Journal of Mathematical Analysis, 49 (4). pp. 3233-3251. ISSN 0036-1410. doi:10.1137/16m1077490. https://resolver.caltech.edu/CaltechAUTHORS:20200331-071759476

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Abstract

We show the existence and uniqueness of a stationary state for a kinetic Fokker--Planck equation modeling the fiber lay-down process in the production of nonwoven textiles. Following a micro-macro decomposition, we use hypocoercivity techniques to show exponential convergence to equilibrium with an explicit rate assuming the conveyor belt moves slowly enough. This work is an extension of [Dolbeault et al., Appl. Math. Res. Express. AMRX, 2013 (2013), pp. 165-175], where the authors consider the case of a stationary conveyor belt. Adding the movement of the belt, the global Gibbs state is not known explicitly. We thus derive a more general hypocoercivity estimate from which existence, uniqueness, and exponential convergence can be derived. To treat the same class of potentials as in [Dolbeault et al., Appl. Math. Res. Express. AMRX, (2013), pp. 165-175] we make use of an additional weight function following the Lyapunov functional approach in [M. Kolb, M. Savov, and A. Wübker, SIAM J. Math. Anal., 45 (2013), pp. 1-13].


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1137/16m1077490DOIArticle
https://arxiv.org/abs/1605.04121arXivDiscussion Paper
ORCID:
AuthorORCID
Hoffmann, Franca0000-0002-1182-5521
Additional Information:© 2017 Society for Industrial and Applied Mathematics. Received by the editors June 22, 2016; accepted for publication (in revised form) June 19, 2017; published electronically August 29, 2017. Funding: The first and third authors acknowledge the support of the ERC grant MATKIT (ERC-2011-StG). The second author acknowledges support from the Engineering and Physical Sciences Research Council (UK) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. EB is very grateful to the University of Cambridge for its sunny hospitality during the second semester of the academic year 2015–2016. The authors are very grateful to the two anonymous referees for very fruitful and detailed comments and remarks.
Funders:
Funding AgencyGrant Number
European Research Council (ERC)MATKIT
Engineering and Physical Sciences Research Council (EPSRC)EP/H023348/1
Subject Keywords:hypocoercivity, rate of convergence, fiber lay-down, existence and uniqueness of a stationary state, perturbation, moving belt
Issue or Number:4
Classification Code:AMS subject classifications: 35B20, 35B40, 35B45, 35Q84
DOI:10.1137/16m1077490
Record Number:CaltechAUTHORS:20200331-071759476
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200331-071759476
Official Citation:Exponential Decay to Equilibrium for a Fiber Lay-Down Process on a Moving Conveyor Belt. Emeric Bouin, Franca Hoffmann, and Clément Mouhot. SIAM Journal on Mathematical Analysis 2017 49:4, 3233-3251; doi: 10.1137/16m1077490
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:102181
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:31 Mar 2020 17:02
Last Modified:16 Nov 2021 18:09

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