A Becker-D{\"o}ring model with injection and irreversible fragmentation

We introduce and analyse a variant of the Becker-D{\"o}ring equations that models the growth of clusters through the gain or loss of monomers. Motivated by enzymatic reactions in biology, this model incorporates irreversible fragmentation and monomers injection. We establish the well-posedness of the equations under suitable conditions on the kinetic rates. Then, as in the Becker-D{\"o}ring equations, we distinguish two cases for the long time behaviour of our solution, however the distinction is made from the constant rate injection of monomers. While under strong fragmentation rate the system may exhibit infinite steady-states, we prove for low injection rate and moderate fragmentation the solution converges locally exponentially fast to the equilibrium. Finally, we present a well-balanced and coarse-grained numerical scheme.