Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel

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.

[1]  M. Smoluchowski,et al.  Drei Vortrage uber Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen , 1916 .

[2]  Tamás Vicsek,et al.  Dynamic Scaling for Aggregation of Clusters , 1984 .

[3]  M. Slemrod,et al.  DYNAMICS OF FIRST ORDER PHASE TRANSITIONS , 1984 .

[4]  Z. A. Melzak.  A scalar transport equation , 1957 .

[5]  Jack Carr,et al.  The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions , 1986 .

[6]  Jack Carr,et al.  Asymptotic behaviour of solutions to the coagulation–fragmentation equations. I. The strong fragmentation case , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[7]  Jack Carr,et al.  Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation , 1994 .

[8]  S. Friedlander,et al.  The self-preserving particle size distribution for coagulation by brownian motion☆ , 1966 .

[9]  Ernst,et al.  Dynamic scaling in the kinetics of clustering. , 1985, Physical review letters.

[10]  T. W. Taylor,et al.  Scaling dynamics of aerosol coagulation. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[11]  Robert M. Ziff,et al.  Coagulation processes with a phase transition , 1984 .

[12]  Robert M. Ziff,et al.  Kinetics of gelation and universality , 1983 .

[13]  Michael Aizenman,et al.  Convergence to equilibrium in a system of reacting polymers , 1979 .

[14]  O. Penrose,et al.  Growth of clusters in a first-order phase transition , 1978 .

[15]  M. H. Ernst,et al.  Scaling solutions of Smoluchowski's coagulation equation , 1988 .

[16]  A. A. Lushnikov.  Evolution of coagulating systems. II. Asymptotic size distributions and analytical properties of generating functions , 1974 .

[17]  D. Lilly,et al.  Solutions to the equations for the kinetics of coagulation , 1965 .

[18]  M. Smoluchowski,et al.  Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen , 1927 .

[19]  P. Hohenberg,et al.  Theory of Dynamic Critical Phenomena , 1977 .

[20]  Exactly soluble addition and condensation models in coagulation kinetics , 1984 .

[21]  E. C. Titchmarsh,et al.  The Laplace Transform , 1991, Heat Transfer 1.

[22]  Warren H. White,et al.  A global existence theorem for Smoluchowski’s coagulation equations , 1980 .

[23]  M. Smoluchowski.  Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen , 1918 .

[24]  I. Lifshitz,et al.  The kinetics of precipitation from supersaturated solid solutions , 1961 .

[25]  Kurt Binder,et al.  Theory for the dynamics of "clusters." II. Critical diffusion in binary systems and the kinetics of phase separation , 1977 .

[26]  I. Stewart.  Density conservation for a coagulation equation , 1991 .