  We consider linear and nonlinear transport equations with irregular velocity
fields, motivated by models coming from mean field games. The velocity fields
are assumed to increase in each coordinate, and the divergence therefore fails
to be absolutely continuous with respect to the Lebesgue measure in general.
For such velocity fields, the well-posedness of first- and second-order linear
transport equations in Lebesgue spaces is established, as well as the existence
and uniqueness of regular ODE and SDE Lagrangian flows. These results are then
applied to the study of certain nonconservative, nonlinear systems of transport
type, which are used to model mean field games in a finite state space. A
notion of weak solution is identified for which a unique minimal and maximal
solution exist, which do not coincide in general. A selection-by-noise result
is established for a relevant example to demonstrate that different types of
noise can select any of the admissible solutions in the vanishing noise limit.
