## Abstract

We provide a method to compute self-similar solutions for various fragmentation equations and use it to compute their asymptotic behaviours. Our procedure is applied to specific cases: (i) the case of mitosis, where fragmentation results into two identical fragments, (ii) fragmentation limited to the formation of sufficiently large fragments, and (iii) processes with fragmentation kernel presenting a power-like behaviour.

## 1. Introduction

Coagulation–fragmentation theory is a very active area of research involving theoretical disciplines such as pure analysis, mathematical physics and probability, and more practical fields among which we can include integral equations, numerical analysis and stochastic processes (random graphs and SPDEs). However, the origins of this theory lie in aerosol physics and a variety of applications have risen even before the first rigorous formal results could be established. Here we provide a tentative list, incomplete due to the extensive development of applications, of representative fields of application integrating references from [1–3] with some more recent study; further references can also be found in [4–6]. Fragmentation processes appear in various scientific contexts such as:

— erosion [9].

— sprays and drop dispersion [10]; breakage of immiscible fluids in turbulent flows [11,12]; fragmentation–coagulation scattering model for the dynamics of stirred liquid–liquid dispersions [13].

— cell cultures, multicellular growing systems and biological tissues [17].

As one can deduce from the literature, the mathematical theory of fragmentation is far more prosperously developed than its coagulation counterpart (a general review is provided in [18]), evidently due to the fact that the theory of linear operators gives general and stronger results. Concerning the study of the *self-similar fragmentation equation*, previous work can be found in the following references: Cheng & Redner [19] give a first general discussion of the kinetics of continuous, irreversible fragmentation processes; Treat in [20] and mainly in [21] settles some fundamental facts about the similarity solutions; Bertoin [22] develops the probabilistic counterpart (fragmentation stochastic processes); Escobedo *et al.* [23], notably, completely solve the existence of self-similar solutions and the convergence problem; later, Michel [24,25] deals with the cell division eigenproblem; Laurençot & Perthame [26] develop bounds for the exponential decay for the growth-fragmentation or cell-division equation; these problems (in particular the growth-fragmentation equations) and the rate of convergence are later considered by Cáceres *et al.* [27] and Balagué *et al.* [28]. The study of fragmentation equations from an evolution semigroup point of view has also been the object of many studies (see [29] and references therein).

### (a) The self-similar equation

We consider here the fragmentation equation with self-similar break-up rate. That is,
*β*(*x*)=*x*^{γ} defines the spontaneous fragmentation rate of *x*-sized clusters and B(*u*) the relative distribution density of the split products. Moreover, B must satisfy the consistency condition

One can use an *invariance argument* to derive the self-similar equation. Consider, in fact, the scale transformations with new variables _{A}(*x*,*t*) given by formula
*A*^{2} preserves total mass of the one-parameter family of solutions, as one can directly check, so that the renormalization group (1.6) provides a formula to generate solutions with different values for the moments M_{β}, _{1} is constant under renormalization (and time evolution), but also all the other moments are conserved. The role of this self-similar function is that of a special physical solution to the fragmentation equation.

The above requirement is a symmetry property that permits determining the self-similar function in an unique way: in this sense, the similarity problem does actually belong to the first kind self-similarity. To do so, we seek a family c_{A} independent of *A*; calling *q*=*Ax* and *p*=*A*^{−γ}(*t*+*t*_{0}), we formally apply the derivative with respect to *A* and get
*γ*:
*C*_{1} and *C*_{2} are free constants and *A* can still be arbitrarily chosen. A possible way to fix *C*_{1} and *C*_{2}, however, is deciding the values of M_{1} and M_{0}(0). Let *Φ* (notice the difference in notation between M_{γ} and *t*_{0}=0 (absence of singularities), straightforward computations lead to
*A*^{2}=`C`_{1}, introduce the self-similar variable *ξ*=*xt*^{1/γ} and relabel *Φ*(*ξ*) in order to get
*self-similar fragmentation equation*:

In this paper, we introduce a new method to integrate the equation for self-similar solutions to fragmentation equations. As a result, we deduce explicit formulae in integral form that are suitable for mathematical analysis. In particular, this allows us to provide sharp asymptotic expansions and regularity results for three different cases: (i) the case when fragmentation results in two identical fragments (mitosis, or with B(*u*) being a Dirac delta function centred in *u*) is finitely supported in an interval excluding *u*=0 and *u*=1, and consists of a combination of Dirac deltas, where we show that it is the fragmentation process producing the smaller fragments that determines the asymptotics of the self-similar solution near the origin, and (iii) the case when B(*u*) behaves as a power law near *u*=0, where we obtain sharp asymptotics and

In §2, we will apply the Mellin transform to (1.13) and deduce a problem defined in the complex plane that will be solved in several cases: the case of mitosis in §3, the case of a function B consisting of a combination of Dirac deltas in §4 and the case of a function B with a power-like behaviour in §5. In appendix A, we provide a detailed estimate of an integral appearing in previous sections.

## 2. Mellin transform and the derived equation

Let *c*(*t*,*x*) be the function representing mass distribution at time *t*. We wish to study its asymptotic behaviour when it obeys the fragmentation equations (1.1) and (1.2). We will study the properties of the self-similar solution *φ*(*ξ*) to the self-similar fragmentation equation:
*β*(*x*)=*x*^{γ}.

Escobedo *et al.* [23] obtained a *Universality theorem for fragmentation without shattering*, which in the case *β*(*ξ*)=*ξ*^{γ}, *γ*>0 and, if for *m*≤*γ*,
*ML*_{m} stands for the space of functions with the *m* moment bounded and *BV* ^{1}(0,1) for the set of functions in *L*^{1}_{loc} such that *φ*_{M} solution to (2.1) with *M*_{1}=M. Moreover, the regularity space for the solution is *s*), which are less regularity and certain asymptotic behaviours near *s*=1 and near *s*=0-, we will obtain estimations, explicit formulae and stronger *φ* in the interval (0,1). Our strategy is to analyse the properties of the Mellin transform *Φ*(*η*) of *φ*(*ξ*), and then deduce from it the relevant information.

We first recall how Mellin transform is defined as
*β*(*ξ*)*φ*(*ξ*),
*s*), which is
*β*(*ξ*)=*ξ*^{γ}, we deduce the following functional problem in the complex plane:

Note that we are interested in evaluating *Φ*(*z*) at *z*=*δ*+*is* in order to apply an inverse Mellin transform. This is defined as
*z*)=*δ* in the complex plane. The reason for this is that *δ* is greater than the real part of all singularities of *Φ*(*z*). This is due to the fact that we request at least that

Much information on the Mellin transform and the so-called Mellin–Barnes integrals can be found in the monograph of Paris & Kaminski [30] to which we refer for details on asymptotic expansions and theoretical results.

## 3. A representation formula for the solution and asymptotics for a mitosis process

We consider here the case of mitosis, that is when particles divide into equally sized fragments. This is represented by
*Ψ*(*z*) satisfies
*z*)<2, is given by
*φ*(*ξ*), we must perform the inverse Mellin transform:
*Φ*(*z*) for ℜ(*z*)<2 combined with the exponential decay as *z*)=*z*_{0}<0. We will choose *z*_{0} as the point of stationary phase, i.e. the solution of
*f*/d*z* and write the following equation for a stationary point *z*_{0}:
*φ*(*ξ*) as *φ*(2*ξ*)≪*φ*(*ξ*), but by a balance between the term at the left-hand side and the first at the right-hand side. Hence,

Another interesting property of the selfsimilar solution, due to the exponential decay of its Mellin transform, is its *n* and hence all derivatives of *φ*(*ξ*) are bounded.

## 4. A representation formula for the solution and asymptotics when B(*u*) is a combination of Dirac deltas

We will study now the effect of a more general distribution of fragmentation processes. Let
*u*_{1}>*u*_{0} and *α*, *β* such that
*αu*_{0} can be viewed as the probability of generating a fragment from an element 1/*u*_{0} its size, while the parameter *βu*_{1} is the probability of generating a fragment from an element 1/*u*_{1} times its size.

Then,
*z* such that ℜ(*z*)<0 (since *u*_{1}/*u*_{0}>1). We can deform the integration contour towards the line ℜ(*z*_{0}) with
*u*_{0} and not on *u*_{1}. Hence, if fragments come from elements of different sizes, then the fragments that are produced by the larger elements (or, equivalently, the fragmentations that produce the fragments with smaller size) will determine the distribution near the origin even if they are less frequent (*αu*_{0}<*βu*_{1}). The arguments above can be extended B(*u*) being the sum of various Dirac deltas or even Dirac deltas and distributions compactly supported away from *u*=0.

## 5. A representation formula for the solution when B(*u*)=*O*(*u*^{μ}), 0<*u*≪1

Let B(*u*) be a continuous function and B(*u*)=*O*(*u*^{μ}) as *μ*>0. Moreover, let *λ* and *c* are constants such that there exist a *ε*>0:
*μ* and *λ* characterize the behaviour of B(*u*).

We introduce

A formal solution to the equation
*S*_{2}. Since
*S*_{2,2} is bounded for all *z* such that ℜ(*z*)>−*μ* and does not grow as *S*_{2,1} consists of the the difference of two generalized harmonic series and, as such, one has
*z*)>−*μ*. The function *ψ*(*s*) is the digamma function, and using its asymptotic expansion we conclude
*S*_{1}, we have
*u*)−B(1)*u*^{μ}−*cu*^{μ}(1−*u*)^{λ}=*O*((1−*u*)^{λ+ε}) as *z* such that ℜ(*z*)>−*μ* and is bounded as *S* is bounded for any *z* such that ℜ(*z*)>−*μ* and
*z*)>−*μ* since
*Φ*(*z*) in the form (5.4) provided ℜ(*z*)>−*μ*. In fact, this expression provides an analytic function for ℜ(*z*)>−*μ*.

By analysing formula (5.4), we can extract important information on the function *φ*(*ξ*). First, we note
*x*=*μ*+*δ*+2, together with
*C* such that

Another property of the self-similar solution is its regularity belonging to the *φ* and derive *n* times:
*i*(*s*/*γ*)+(*x*/*γ*)|^{n}|*Γ*(*i*(*s*/*γ*)+*x*/*γ*)| is bounded by exponentially decaying functions of *s*. Hence we conclude that, for *x*>−*μ*+*n*,

Note that the function B(*u*) only enters into the analysis through the integral (5.5). One could relax the continuity hypothesis for B(*u*) and consider just integrability in an interval away from *u*=0 and *u*=1, to obtain identical results concerning asymptotic behaviour and differentiability of *φ*(*ξ*).

## Authors' contributions

Both authors contributed equally to this work.

## Competing interests

We declare we have no competing interests.

## Funding

This work has been supported by grant MTM2014-57158-R from the Spanish Government.

## Acknowledgements

The authors thank J. A. Cañizo for stimulating conversation on the topic of this article.

## Appendix A. Estimate of integrals

In this appendix, we estimate the integral
*f*(*z*) is analytic in a ball of radius *z*_{0}. We split the integral in a region of radius *ε*>0 sufficiently small), where the integrand is analytic, and the region |*z*−*z*_{0}|>*R*, so that
*n*≥3,

- Received September 29, 2016.
- Accepted April 10, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.