# American Institute of Mathematical Sciences

June  2020, 9(2): 431-446. doi: 10.3934/eect.2020012

## Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation

 1 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India 2 Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore-560065, Karnataka, India

Received  March 2019 Revised  April 2019 Published  August 2019

In this paper we study the continuous coagulation and multiple fragmentation equation for the mean-field description of a system of particles taking into account the combined effect of the coagulation and the fragmentation processes in which a system of particles growing by successive mergers to form a bigger one and a larger particle splits into a finite number of smaller pieces. We demonstrate the global existence of mass-conserving weak solutions for a wide class of coagulation rate, selection rate and breakage function. Here, both the breakage function and the coagulation rate may have algebraic singularity on both the coordinate axes. The proof of the existence result is based on a weak $L^1$ compactness method for two different suitable approximations to the original problem, namely, the conservative and non-conservative approximations. Moreover, the mass-conservation property of solutions is established for both approximations.

Citation: Prasanta Kumar Barik. Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation. Evolution Equations & Control Theory, 2020, 9 (2) : 431-446. doi: 10.3934/eect.2020012
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