An Application of Semigroup Theory to a Fragmentation Equation

Abstract

The process of fragmentation arises in many physical situations, including polymer degradation, droplet breakage and rock crushing and grinding. Under suitable assumptions, the evolution of the size distribution c(x, t), where x represents particle size and t is time, may be described by the linear integro-differential equation

$$ \frac{{\partial c(x,t)}} {{\partial t}} = - a(x)c(x,t) + \int_x^\infty {a(y)b(x,y)c(y,t)dy,x > 0,t > 0;} $$

(4.1)

see [1] for details. In (4.1), a(x) describes the rate at which a particle of size x fragments and b(x, y) represents the rate of production of particles of size x due to the break-up of particles of size y > x. For the total mass in the system to remain constant during fragmentation, b must satisfy the condition

$$ \int_0^y {xb(x,y)dx = y.} $$