logo AFST
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.

Reçu le : 2007-10-17
Accepté le : 2008-09-24
Publié le : 2010-12-06
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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 17},
     number = {3},
     year = {2008},
     pages = {461-493},
     doi = {10.5802/afst.1190},
     zbl = {pre05505536},
     mrnumber = {2488229},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2008_6_17_3_461_0/}
}
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/item/AFST_2008_6_17_3_461_0/

[1] Aronson (D. G.).— The fundamental solution of a linear parabolic equation containing a small parameter. Ill. J. Math. 3, 580-619 (1959). | MR 107758 | Zbl 0090.07601

[2] Ball (J. M.), Carr (J.), Penrose (O.).— The Becker-Döring cluster equations : basic properties and asymptotic behaviour of solutions, Comm. Math. Phys. 104, 657-692 (1986). | MR 841675 | Zbl 0594.58063

[3] Ball (J. M.), Carr (J.).— Asymptotic behaviour of solutions to the Becker-Döring equations for arbitrary initial data. Proc. Royal Soc. Edin. Sect. A 108, 109-116 (1988). | MR 931012 | Zbl 0656.58021

[4] Benilan (P.), Wrzosek (D.).— On an infinite system of reaction-diffusion equations. Adv. Math. Sci. Appl. 7, 351-366 (1997). | MR 1454671 | Zbl 0884.35165

[5] Chae (D.), Dubovski (P. B.).— Existence and uniqueness for spatially inhomogeneous coagulation equation with sources and effluxes. Z. angew Math. Phys. (ZAMP) 46, 580-594 (1995). | MR 1345813 | Zbl 0833.35142

[6] Collet (J. F.), Poupaud (F.).— Existence of solutions to coagulation-fragmentation systems with diffusion. Transport Theory Stat. Phys. 25, 503-513 (1996). | MR 1407550 | Zbl 0870.35117

[7] Dubovski (P. B.).— Mathematical Theory of Coagulation. Seoul National University, Research Institute of Mathematics, 1994. | MR 1290321 | Zbl 0880.35124

[8] Dubovski (P. B.).— Solubility of the transport equation arising in the kinetics of coagulation and fragmentation. Izvestiya : Mathematics 65, 1-22 (2001). | MR 1829401 | Zbl 1097.82024

[9] Friedman (A.).— Partial differential equations of parabolic type. Prentice Hall 1964. | MR 181836 | Zbl 0144.34903

[10] Galkin (V. A.).— On stability and stabilization of solutions of the coagulation equation. Differential Equations 14, 1863-1874 (1978). | MR 515104 | Zbl 0409.45011

[11] Galkin (V. A.).— Generalized solution of the Smoluchowski kinetic equation for spatially inhomogeneous systems. Dokl. Akad. Nauk SSSR 293, 74-77 (1987). | MR 882081

[12] Ha (S.-Y.), Tzavaras (A.).— Lyapunov functionals and L 1 -stability for discrete velocity Boltzmann equations. Comm. Math. Phys. 239, 65-92 (2003). | MR 1997116 | Zbl 1024.35067

[13] Ha (S.-Y.).— L 1 stability estimate for a one-dimensional Boltzmann equation with inelastic collisions. J. Differential Equations 190, 621-642 (2003). | MR 1971148 | Zbl pre01929526

[14] Laurencot (P.), Wrzosek (D.).— The Becker-Döring model with diffusion. I. Basic properties of solutions. Colloq. Math. 75, 245-269 (1998). | MR 1490692 | Zbl 0894.35055

[15] Liu (T.-P.), Yang (T.).— Well-posedness theory for hyperbolic conservation laws. Comm. Pure Appl. Math. 52, 1553-1586 (1999). | MR 1711037 | Zbl 1034.35073

[16] Penrose (O.), Lebowitz (J. L.).— Towards a rigirous molecular theory of metastability, in Studies in Statistical Mechanics VII (Fluctuation Phenomena). E. Montroll and J.L.Lebowitz, eds. North-Holland, Amsterdam, 293-340 1979.

[17] Slemrod (M.).— Trend to equilibrium in the Becker-Döring cluster equations. Nonlinearity 2, 429-443 (1989). | MR 1005058 | Zbl 0709.60528

[18] Slemrod (M.), Grinfeld (M.), Qi (A.), Stewart (I.).— A discrete velocity coagulation-fragmentation model. Math. Meth. Appl. Sci. 18, 959-994 (1995). | MR 1350710 | Zbl 0834.76080

[19] Slemrod (M.).— Metastable fluid flow described via a discrete-velocity coagulation–fragmentation model. J. Stat. Phys. 83, 1067-1108 (1996). | MR 1392420 | Zbl 1081.82621

[20] Voloschuk (V. M.), Sedunov (Y. S.).— Coagulation Processes in Disperse Systems. Leningrad : Gidrometeoizdat, 1975 (in Russian).

[21] Wrzosek (D.).— Existence of solutions for the discrete coagulation–fragmentation model with diffusion. Topol. Methods Nonlinear Analysis 9, 279-296 (1997). | MR 1491848 | Zbl 0892.35077