Dynamics of droplets in an agitated dispersion with multiple breakage. Part II: uniqueness and global existence Antonio Fasano, Fabio Rosso Mathematics Subject Classification (1991): 82C99,76T20,76T99,45K05 This work was partially supported by the G.N.F.M.–I.N.D.A.M. Strategic Project “Metodi Matematici in Fluidodinamica e Dinamica Molecolare” Abstract. In Part I (see [2]) a new model for the evolution of a system of droplets dispersed in an agitated liquid was presented. Our aim was to extend a previous version (see [4]) in order to describe the influence of each breakage mode. Here we complete the mathematical analysis to ensure the well posedness (in the sense of Hadamard) of the Cauchy problem for the main evolution equation. 1. Introduction In Part I of this paper (see [2] this same volume) we presented a very general model for spatially homogeneous liquid dispersions in which any possible rupture mode is considered and the corresponding breakage frequencies are allowed to blow up as the droplet volume approaches a critical finite upper bound. Namely we allow a parent droplet to break in at most N pieces where N can be any finite positive integer greater or equal than two. The breakage frequency α k of the k−th mode is allowed to tend to infinity as v tends to v m (as in [1]). The main purpose of Part I was to show how to deal with the probability functions for each breakage mode and to verify the physical consistency of the whole model. Such a model generalized the one proposed in [4] (limited to binary breakage and with bounded breakage frequency), where the so–called volume scattering operator was introduced, preventing the appearance of droplets beyond a critical size (depending on the agitation speed). Here we prove the well posedness (in the sense of Hadamard) of the Cauchy problem for the evolution equation derived in Part I.