On General Transport Equations with Abstract Boundary Conditions.The Case of Divergence Free Force Field

This work is a continuation of our previous paper [8]. We investigate well-posedness (in the semigroup theory sense) of transport equations with general external fields and general measures associated to boundary conditions modeled by abstract boundary operators H. Fine properties of the traces are investigated, extending well-known results by M. Cessenat. For dissipative boundary conditions, we revisit and generalize known results while, for multiplicative boundary conditions we extend techniques from a recent contribution of B. Lods. Finally, we also investigate the case of boundary conditions associated to a boundary operator of norm one, extending recent results to more general fields and measures.