Existence, uniqueness and stability for spatially inhomogeneous Becker-Döring equations with diffusion and convection terms

P. B. Dubovski[1]; S.-Y. Ha[2]

  • [1] Department of Mathematical Sciences, Stevens Institute of Technology.
  • [2] Department of Mathematical Sciences, Seoul National University.

Annales de la faculté des sciences de Toulouse Mathématiques (2008)

  • Volume: 17, Issue: 3, page 461-493
  • ISSN: 0240-2963

Abstract

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We consider the spatially inhomogeneous Bekker-Döring infinite-dimensional kinetic system describing the evolution of coagulating and fragmenting particles under the influence of convection and diffusion. The simultaneous consideration of opposite coagulating and fragmenting processes causes many additional difficulties in the investigation of spatially inhomogeneous problems, where the space variable changes differently for distinct particle sizes. To overcome these difficulties, we use a modified maximum principle and establishes the local-in-time existence and uniqueness of continuous solutions for unbounded kinetic coefficients that allow their linear growth. The global-in-time existence, uniqueness, and stability theorems for classical solutions are also obtained for bounded kinetic coefficients, and these are based on a new trick, which enables to obtain new a priori estimates for classical solutions regardless of the above mentioned non-uniform change of the spatial variable in the distribution function. We also show that the solutions are stable with respect to small perturbations in l 1 of both initial data and kinetic coefficients. Our methods allow to treat zero diffusion coefficients limit for some sizes of the particles and, moreover, can be employed to prove the vanishing diffusion limit that the solution of the system with diffusion approaches to the solution of the system with the transport terms only. We establish the uniform stability theorems in L 1 for purely coagulating or purely fragmenting kinetic systems. This new stability result is based on the explicit construction of robust Lyapunov functionals and their decay estimates in time.

How to cite

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Dubovski, P. B., and Ha, S.-Y.. "Existence, uniqueness and stability for spatially inhomogeneous Becker-Döring equations with diffusion and convection terms." Annales de la faculté des sciences de Toulouse Mathématiques 17.3 (2008): 461-493. <http://eudml.org/doc/10093>.

@article{Dubovski2008,
abstract = {We consider the spatially inhomogeneous Bekker-Döring infinite-dimensional kinetic system describing the evolution of coagulating and fragmenting particles under the influence of convection and diffusion. The simultaneous consideration of opposite coagulating and fragmenting processes causes many additional difficulties in the investigation of spatially inhomogeneous problems, where the space variable changes differently for distinct particle sizes. To overcome these difficulties, we use a modified maximum principle and establishes the local-in-time existence and uniqueness of continuous solutions for unbounded kinetic coefficients that allow their linear growth. The global-in-time existence, uniqueness, and stability theorems for classical solutions are also obtained for bounded kinetic coefficients, and these are based on a new trick, which enables to obtain new a priori estimates for classical solutions regardless of the above mentioned non-uniform change of the spatial variable in the distribution function. We also show that the solutions are stable with respect to small perturbations in $l^1$ of both initial data and kinetic coefficients. Our methods allow to treat zero diffusion coefficients limit for some sizes of the particles and, moreover, can be employed to prove the vanishing diffusion limit that the solution of the system with diffusion approaches to the solution of the system with the transport terms only. We establish the uniform stability theorems in $L^1$ for purely coagulating or purely fragmenting kinetic systems. This new stability result is based on the explicit construction of robust Lyapunov functionals and their decay estimates in time.},
affiliation = {Department of Mathematical Sciences, Stevens Institute of Technology.; Department of Mathematical Sciences, Seoul National University.},
author = {Dubovski, P. B., Ha, S.-Y.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {6},
number = {3},
pages = {461-493},
publisher = {Université Paul Sabatier, Toulouse},
title = {Existence, uniqueness and stability for spatially inhomogeneous Becker-Döring equations with diffusion and convection terms},
url = {http://eudml.org/doc/10093},
volume = {17},
year = {2008},
}

TY - JOUR
AU - Dubovski, P. B.
AU - Ha, S.-Y.
TI - Existence, uniqueness and stability for spatially inhomogeneous Becker-Döring equations with diffusion and convection terms
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 3
SP - 461
EP - 493
AB - We consider the spatially inhomogeneous Bekker-Döring infinite-dimensional kinetic system describing the evolution of coagulating and fragmenting particles under the influence of convection and diffusion. The simultaneous consideration of opposite coagulating and fragmenting processes causes many additional difficulties in the investigation of spatially inhomogeneous problems, where the space variable changes differently for distinct particle sizes. To overcome these difficulties, we use a modified maximum principle and establishes the local-in-time existence and uniqueness of continuous solutions for unbounded kinetic coefficients that allow their linear growth. The global-in-time existence, uniqueness, and stability theorems for classical solutions are also obtained for bounded kinetic coefficients, and these are based on a new trick, which enables to obtain new a priori estimates for classical solutions regardless of the above mentioned non-uniform change of the spatial variable in the distribution function. We also show that the solutions are stable with respect to small perturbations in $l^1$ of both initial data and kinetic coefficients. Our methods allow to treat zero diffusion coefficients limit for some sizes of the particles and, moreover, can be employed to prove the vanishing diffusion limit that the solution of the system with diffusion approaches to the solution of the system with the transport terms only. We establish the uniform stability theorems in $L^1$ for purely coagulating or purely fragmenting kinetic systems. This new stability result is based on the explicit construction of robust Lyapunov functionals and their decay estimates in time.
LA - eng
UR - http://eudml.org/doc/10093
ER -

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