Estimating the division rate and kernel in the fragmentation equation - Archive ouverte HAL Access content directly
Journal Articles Annales de l'Institut Henri Poincaré C, Analyse non linéaire Year : 2018

## Estimating the division rate and kernel in the fragmentation equation

Marie Doumic
Miguel Escobedo
• Function : Author
• PersonId : 856874
Magali Tournus
• Function : Author
• PersonId : 962772

#### Abstract

We consider the fragmentation equation $\dfrac{\partial}{\partial t}f (t, x) = −B(x)f (t, x) + \int_{ y=x}^{ y=\infty} k(y, x)B(y)f (t, y)dy,$ and address the question of estimating the fragmentation parameters-i.e. the division rate $B(x)$ and the fragmentation kernel $k(y, x)$-from measurements of the size distribution $f (t, ·)$ at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance (Xue, Radford, Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a polynomial division rate $B(x) = \alpha x^{\gamma}$ and a self-similar fragmentation kernel $k(y, x) = \frac{1}{y} k_0 (x/ y)$, we use the asymptotic behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to obtain uniqueness of the triplet $(\alpha, \gamma, k _0)$ and a representation formula for $k_0$. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.

#### Domains

Mathematics [math] Analysis of PDEs [math.AP]

### Dates and versions

hal-01501811 , version 1 (04-04-2017)
hal-01501811 , version 2 (24-04-2018)

### Identifiers

• HAL Id : hal-01501811 , version 2
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• DOI :

### Cite

Marie Doumic, Miguel Escobedo, Magali Tournus. Estimating the division rate and kernel in the fragmentation equation. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2018, 35 (7), pp.1847-1884. ⟨10.1016/j.anihpc.2018.03.004⟩. ⟨hal-01501811v2⟩

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