  We study first- and second-order linear transport equations, as well as ODE
and SDE flows, with velocity fields satisfying a one-sided Lipschitz condition.
Depending on the time direction, the flows are either compressive or expansive.
In the compressive regime, we characterize the stable continuous distributional
solutions of both the first and second-order nonconservative transport
equations as the unique viscosity solution. Our results in the expansive regime
complement the theory of Bouchut, James, and Mancini, and we provide a complete
theory for both the conservative and nonconservative equations in Lebesgue
spaces, as well as proving the existence, uniqueness, and stability of the
regular Lagrangian ODE flow. We also provide analogous results in this context
for second order equations and SDEs with degenerate noise coefficients that are
constant in the spatial variable.
