Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel
Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel
复制标题
具有常数核的 Smoluchowski 凝固方程的动态标度证明
DOI：
10.1007/bf02186868
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发表时间：
1994
影响因子：
1.6
通讯作者：
O. Penrose
中科院分区：
文献类型：
--
作者：
M. Kreer;O. Penrose
AbstractSmoluchowski's coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version wherect=c1(t) is the concentration ofl-particle clusters at timet. We prove that for initial data satisfyingc1(0)>0 and the condition 0 ⩽cl(0) 0), the solutions behave asymptotically likec1(t)∼t−2≈c(lt−1) ast→∞ withlt−1 kept fixed. The scaling function ≈c(ξ) is (1/gr)ξ, where
$$\rho = \sum _l lc_l (0)$$
, a conserved quantity, is the initial number of particles per unit volume. An analous result is obtained for the continuous version of Smoluchowski's coagulation equation
$$\frac{\partial }{{\partial t}}c(v,{\text{ }}t) = \int_0^v {du{\text{ }}c(v - u,{\text{ }}t){\text{ }}c(u,{\text{ }}t) - 2c(v,{\text{ }}t)} \int_0^\infty {du{\text{ }}c(u,{\text{ }}t)}$$
wherec(v, t) is the oncentration of clusters of sizev.
AbstractSmoluchowski's coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version wherect=c1(t) is the concentration ofl-particle clusters at timet. We prove that for initial data satisfyingc1(0)>0 and the condition 0 ⩽cl(0) 0), the solutions behave asymptotically likec1(t)∼t−2≈c(lt−1) ast→∞ withlt−1 kept fixed. The scaling function ≈c(ξ) is (1/gr)ξ, where
$$\rho = \sum _l lc_l (0)$$
, a conserved quantity, is the initial number of particles per unit volume. An analous result is obtained for the continuous version of Smoluchowski's coagulation equation
$$\frac{\partial }{{\partial t}}c(v,{\text{ }}t) = \int_0^v {du{\text{ }}c(v - u,{\text{ }}t){\text{ }}c(u,{\text{ }}t) - 2c(v,{\text{ }}t)} \int_0^\infty {du{\text{ }}c(u,{\text{ }}t)}$$
wherec(v, t) is the oncentration of clusters of sizev.