### Abstract

is treated as an abstract Cauchy problem, posed in an appropriate Banach space. Perturbation techniques from the theory of semigroups of operators are used to establish the existence and uniqueness of physically meaningful solutions under minimal restrictions on the fragmentation rates. In one particular case an explicit formula for the associated semigroup is obtained and this enables additional properties, such as compactness of the resolvent and analyticity of the semigroup, to be deduced. Another explicit solution of this particular fragmentation problem, in which mass is apparently created from a zero-mass initial state, is also investigated, and the theory of Sobolev towers is used to prove that the solution actually emanates from an initial infinite cluster of unit mass.

Language | English |
---|---|

Pages | 181-201 |

Number of pages | 21 |

Journal | Journal of Evolution Equations |

Volume | 12 |

Issue number | 1 |

Early online date | 26 Nov 2011 |

DOIs | |

Publication status | Published - Mar 2012 |

### Fingerprint

### Keywords

- semigroup
- discrete fragmentation
- Sobolev tower

### Cite this

*Journal of Evolution Equations*,

*12*(1), 181-201. https://doi.org/10.1007/s00028-011-0129-8

}

*Journal of Evolution Equations*, vol. 12, no. 1, pp. 181-201. https://doi.org/10.1007/s00028-011-0129-8

**Discrete fragmentation with mass loss.** / Smith, Ann Louise; Lamb, Wilson; Langer, Matthias; McBride, Adam.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Discrete fragmentation with mass loss

AU - Smith, Ann Louise

AU - Lamb, Wilson

AU - Langer, Matthias

AU - McBride, Adam

N1 - Added pdf document and references

PY - 2012/3

Y1 - 2012/3

N2 - We examine an infinite system of ordinary differential equations that models a discrete fragmentation process in which mass loss can occur. The problemis treated as an abstract Cauchy problem, posed in an appropriate Banach space. Perturbation techniques from the theory of semigroups of operators are used to establish the existence and uniqueness of physically meaningful solutions under minimal restrictions on the fragmentation rates. In one particular case an explicit formula for the associated semigroup is obtained and this enables additional properties, such as compactness of the resolvent and analyticity of the semigroup, to be deduced. Another explicit solution of this particular fragmentation problem, in which mass is apparently created from a zero-mass initial state, is also investigated, and the theory of Sobolev towers is used to prove that the solution actually emanates from an initial infinite cluster of unit mass.

AB - We examine an infinite system of ordinary differential equations that models a discrete fragmentation process in which mass loss can occur. The problemis treated as an abstract Cauchy problem, posed in an appropriate Banach space. Perturbation techniques from the theory of semigroups of operators are used to establish the existence and uniqueness of physically meaningful solutions under minimal restrictions on the fragmentation rates. In one particular case an explicit formula for the associated semigroup is obtained and this enables additional properties, such as compactness of the resolvent and analyticity of the semigroup, to be deduced. Another explicit solution of this particular fragmentation problem, in which mass is apparently created from a zero-mass initial state, is also investigated, and the theory of Sobolev towers is used to prove that the solution actually emanates from an initial infinite cluster of unit mass.

KW - semigroup

KW - discrete fragmentation

KW - Sobolev tower

UR - http://www.scopus.com/inward/record.url?scp=84864117636&partnerID=8YFLogxK

U2 - 10.1007/s00028-011-0129-8

DO - 10.1007/s00028-011-0129-8

M3 - Article

VL - 12

SP - 181

EP - 201

JO - Journal of Evolution Equations

T2 - Journal of Evolution Equations

JF - Journal of Evolution Equations

SN - 1424-3199

IS - 1

ER -