Existence, uniqueness and stability for spatially inhomogeneous Becker-Döring equations with diffusion and convection terms
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 3, pp. 461-493.

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.

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.

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DOI : https://doi.org/10.5802/afst.1190
@article{AFST_2008_6_17_3_461_0,
author = {P. B. Dubovski and S.-Y. Ha},
title = {Existence, uniqueness and stability for spatially inhomogeneous {Becker-D\"oring} equations with diffusion and convection terms},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {461--493},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 17},
number = {3},
year = {2008},
doi = {10.5802/afst.1190},
zbl = {pre05505536},
mrnumber = {2488229},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1190/}
}
P. B. Dubovski; S.-Y. Ha. Existence, uniqueness and stability for spatially inhomogeneous Becker-Döring equations with diffusion and convection terms. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 3, pp. 461-493. doi : 10.5802/afst.1190. https://afst.centre-mersenne.org/articles/10.5802/afst.1190/

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