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].
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| 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. | |||||||||
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| 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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