# An Application of Semigroup Theory to a Fragmentation Equation

Chapter

## 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 see [1] for details. In (4.1),

*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)

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

## Keywords

Semigroup Theory Fragmentation Equation Abstract Cauchy Problem Perturbation Theorem Exte Nsion
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## References

- 1.E.D. McGrady and R.M. Ziff, “Shattering” transition in fragmentation,
*Phys. Rev. Lett.***58**(1987), 892–895.MathSciNetCrossRefGoogle Scholar - 2.K-J. Engel and R. Nagel,
*One-parameter semigroups for linear evolution equations*, Springer-Verlag, New York, 1999.Google Scholar - 3.W. Lamb and A.C. McBride, On a continuous coagulation and fragmentation equation with a singular fragmentation kernel, in
*Recent contributions to evolution equations*, Lect. Notes, Marcel Dekker (to appear).Google Scholar - 4.D.J. McLaughlin, W. Lamb, and A.C. McBride, A semigroup approach to fragmentation models,
*SIAM J. Math. Anal.***28**(1997), 1158–1172.MathSciNetCrossRefMATHGoogle Scholar - 5.J. Banasiak, On an extension of the Kato-Voigt perturbation theorem for substochastic semigroups and its application,
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