Strong fragmentation and coagulation with power-law rates

Abstract

Existence of global classical solutions to fragmentation and coagulation equations with unbounded coagulation rates has been recently proved for initial conditions with finite higher order moments. These results cannot be directly generalized to the most natural space of solutions with finite mass and number of particles due to the lack of precise characterization of the domain of the generator of the fragmentation semigroup. In this paper we show that such a generalization is possible in the case when both fragmentation and coagulation are described by power-law rates which are commonly used in the engineering practice. This is achieved through direct estimates of the resolvent of the fragmentation operator, which in this case is explicitly known, proving that it is sectorial and carefully intertwining the corresponding intermediate spaces with appropriate weighted L1 spaces

Analytic Fragmentation Semigroups and Classical Solutions to Coagulation–fragmentation Equations — a Survey

“…This prompted interest in the topic and, subsequently, it was shown that a class of fragmentation operators, that includes physically relevant binary and homogeneous fragmentations, both in the discrete and continuous case, is sectorial albeit in a smaller space of densities that have finite higher moments (the space l 1 p := l 1 U with U = i p , p > 1, or its equivalent X 0,p = L 1 (R + , (1 + x p )dx) in the continuous case), [6,11]. Moreover, for the so-called power law fragmentation it was proven that analyticity holds also in the basic space X 0,1 , [14]. These results also allowed for an explicit characterization of the domains of the fragmentation operator.…”

“…We note that (32) cannot hold for p = 1 as ∆ (1) = 0. Nevertheless, there are analytic fragmentation semigroups in l 1 1 and X 0,1 , see [14,60], but the proofs in these spaces require direct estimates of the resolvent that is not explicitly available in general.…”

Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients

“…This has changed in the recent few years with the realization that the fragmentation semigroup is analytic for a large class of fragmentation rates a and the daughter distribution functions b. This, in turn, allowed for proving the classical solvability of (1) even if the coagulation rate k is unbounded as long as it is dominated in a suitable sense by the fragmentation rate a, see [7,8] and [5,Section 8.1.2]. The proofs use interpolation spaces between X 0,m and the domain of the generator of the fragmentation semigroup in this space.…”

The continuous coagulation equation with multiple fragmentation

“…The main novelty of this article is to show the existence, uniqueness and mass-conserving property of global solutions to (1.1)-(1.2) for singular coagulation and breakup kernels alongwith the quadratic growth on collision kernels. The existence and uniqueness of solutions to the continuous coagulation with linear fragmentation processes have been discussed in many articles by using various techniques for different growth conditions on coagulation and fragmentation rates, see [3,4,5,6,12,19,23,27,29,30,33,34]. However, the nonlinear fragmentation equation has not been extensively addressed.…”

“…Several researchers have already discussed the existence of global classical solutions to the coagulation and linear fragmentation equations through various techniques, see [3,4,5,6,16,22,27,29,30,32]. In [3,4,5,6,27,29,30], authors have discussed the existence of global classical solutions for the coagulation and linear fragmentation equations by using semigroup technique whereas in [16,22,32], a different approach, introduced by Galkin and Dubovskii, is used to show the existence of global classical solutions which relies on a compactness argument. In [22], they have discussed the existence and uniqueness of global solutions to pure coagulation equation by taking an account of unbounded coagulation kernels.…”