Contractivity for Smoluchowski's coagulation equation with solvable kernels

Abstract

We show that the Smoluchowski coagulation equation with the solvable kernels K(x,y) equal to 2, x+y or xy is contractive in suitable Laplace norms. In particular, this proves exponential convergence to a self‐similar profile in these norms. These results are parallel to similar properties of Maxwell models for Boltzmann‐type equations, and extend already existing results on exponential convergence to self‐similarity for Smoluchowski's coagulation equation.

Coalescense with arbitrary-parameter kernels and monodisperse initial conditions: A study within combinatorial framework

“…In this Section, we will compare theoretical solutions arising from the combinatorial framework which were derived in Section III to numerical results obtained by the simulation. 42)- (44). Although theoretical results are defined only for integer s, solid lines are used as guidelines for eyes.…”

“…Taking advantage of a known relation for Bell polynomials [74], (46) we can simplify the set of Eqs. ( 42)- (44) to the form of…”

“…Nevertheless, considerable literature exists on the existence and uniqueness of solutions to some general classes of kernels in Smoluchowski approach, including self-similar dynamical scaling solutions [36][37][38][39]. This area is still exhaustively researched [40][41][42][43][44]. On the basis of the Smoluchowski equation and the works of Ziff [33], a kinetic model of the gelling system consisting of two types of monomers was recently investigated [28].…”

“…Solid lines represent theoretical predictions based on the combinatorial equations, Eqs. (42)-(44). Although theoretical results are defined only for integer s, solid lines are used as guidelines for eyes.…”