Strong fragmentation and coagulation with power-law rates

Jacek Banasiak,Matthias Langer,Wilson Lamb

doi:10.1007/s10665-012-9596-3

Abstract

Existence of global classical solutions to fragmentation and coagulation equations with unbounded coagulation rates has been recently proved for initial conditions with finite higher order moments. These results cannot be directly generalized to the most natural space of solutions with finite mass and number of particles due to the lack of precise characterization of the domain of the generator of the fragmentation semigroup. In this paper we show that such a generalization is possible in the case when both fragmentation and coagulation are described by power-law rates which are commonly used in the engineering practice. This is achieved through direct estimates of the resolvent of the fragmentation operator, which in this case is explicitly known, proving that it is sectorial and carefully intertwining the corresponding intermediate spaces with appropriate weighted L1 spaces.

Highlights

Coagulation and fragmentation processes can be found in many important areas of science and engineering
In recent papers [24,25], we have proved the existence of global classical solutions to the fragmentation–coagulation equation with unbounded fragmentation and coagulation rates
By restricting attention to fragmentation rates of power-law type, we have been able to prove that the related fragmentation semigroup is analytic in the physically relevant space Y = L1(R+, x dx)

Summary

Introduction

Coagulation and fragmentation processes can be found in many important areas of science and engineering. Examples range from astrophysics [1], blood clotting [2], colloidal chemistry and polymer science [3,4] to molecular beam epitaxy [5]. An efficient way of modelling the dynamical behaviour of such processes is to use a rate equation. This paper is dedicated to Professor P.

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