"The initial-boundary value problem for the Lifshitz-Slyozov system with inflow boundary conditions: Analysis and applications"
The Lifshitz-Slyozov system describes the temporal evolution of a mixture of particles ('atoms') and aggregates, where individual atoms can attach to or detach from already existing clusters. The aggregate distribution follows a transport equation with respect to a size variable, whose transport rates are coupled to the dynamic of atoms through a mass conservation relation. Being a system traditionally designed to model phase transitions, the attachment and detachment rates proposed by Lifshitz and Slyozov are such that no boundary condition at zero size is needed. However, the scope of this model is becoming wider (e.g. descriptions of protein polymerization or tentative applications to oceanography). These situations impose attachment and detachment rates that requiere a boundary condition at zero size, which is intrepreted as the synthesis of new clusters from atoms by a nucleation process. Up to date, the mathematical results on this new setting are scarce. In this contribution we study existence and uniquenes of local-in-time solutions when nonlinear boundary conditions are used, together with continuation criteria and results on long-time behavior. We are able to deal with attachment and detachment rates that may eventually lack Lipschitz regularity, like power-law rates. This requires a careful analysis of the characteristic curves associated to the transport process.