Estimating the division rate and kernel in the fragmentation equation

  • Marie Doumic

    INRIA Rocquencourt, équipe-projet MAMBA, domaine de Voluceau, BP 105, 78153 Rocquencourt, France
  • Miguel Escobedo

    Universidad del País Vasco, Facultad de Ciencias y Tecnología, Departamento de Matemáticas, Barrio Sarriena s/n 48940 Lejona (Vizcaya), Spain
  • Magali Tournus

    Centrale Marseille, I2M, UMR 7373, CNRS, Aix-Marseille univ., Marseille, 13453, France
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Abstract

We consider the fragmentation equation

ft(t,x)=B(x)f(t,x)+y=xy=k(y,x)B(y)f(t,y)dy,\frac{\partial f}{\partial t}(t,x) = −B(x)f(t,x) + \int \limits_{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)B(x) and the fragmentation kernel k(y,x)k(y,x) – from measurements of the size distribution f(t,)f(t, \cdot ) 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 and Radford (2013) [26] for amyloid fibril breakage. Under the assumption of a polynomial division rate B(x)=αxγB(x) = \alpha x^{\gamma } and a self-similar fragmentation kernel k(y,x)=1yk0(xy)k(y,x) = \frac{1}{y}k_{0}(\frac{x}{y}), we use the asymptotic behavior proved in Escobedo et al. (2004) [11] to obtain uniqueness of the triplet (α,γ,k0)(\alpha ,\gamma ,k_{0}) and a representation formula for k0k_{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.

Cite this article

Marie Doumic, Miguel Escobedo, Magali Tournus, Estimating the division rate and kernel in the fragmentation equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 7, pp. 1847–1884

DOI 10.1016/J.ANIHPC.2018.03.004