Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients

,Jacek Banasiak,,Institute Of Mathematics, Łódź University Of Technology, Łódź, Poland

doi:10.3934/dcdss.2020161

Abstract

In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.

Highlights

Coagulation equations, introduced by Smoluchowski [22, 23] in the discrete case and in [21] in the continuous one, and extended in [9, 12, 20, 19, 26] to include the reverse fragmentation processes, have proved crucial in numerous applications, ranging from polymerization, aerosol formation, animal groupings, phytoplankton dynamics, to rock crushing and planetesimals formation, see a survey in [5, Vol I] and, as such, they have been extensively studied in engineering, physical and mathematical sciences
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
The only other consequence of the generation theorem of [7] is that (GF0,m (t))t≥0 is a quasi-contractive semigroup (that is, satisfying GF0,m L(X0,m) ≤ eωmt for some ωm) which in turn allowed in the proof of [7, Theorem 2.2] to use the Trotter–Kato representation formula to prove that certain auxiliary semigroups are positive

Summary

Introduction

Coagulation equations, introduced by Smoluchowski [22, 23] in the discrete case and in [21] in the continuous one, and extended in [9, 12, 20, 19, 26] to include the reverse fragmentation processes, have proved crucial in numerous applications, ranging from polymerization, aerosol formation, animal groupings, phytoplankton dynamics, to rock crushing and planetesimals formation, see a survey in [5, Vol I] and, as such, they have been extensively studied in engineering, physical and mathematical sciences. The operator method provides the existence of unique, mass conserving and classical solutions but, while being able to deal with even very singular fragmentation processes, its applications to the full problem (1) for a long time were restricted to bounded coagulation kernels. 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. We show that if the coefficients of (1) satisfy the assumptions of the local existence theorem, the solutions to the truncated problems, constructed in e.g. [16] as the approximations to a weak solution to (1) in the weak compactness method, converge strongly to the classical solution of (1) on its maximal interval of existence, confirming the fact, not entirely surprising, that both methods agree with each other whenever they are both applicable

Hence δ

Using the fact that