  We establish existence and uniqueness results for initial-boundary value
problems for transport equations in one space dimension with nearly
incompressible velocity fields, under the sole assumption that the fields are
bounded. In the case where the velocity field is either nonnegative or
nonpositive, one can rely on similar techniques as in the case of the Cauchy
problem. Conversely, in the general case we introduce a new and more
technically demanding construction, which heuristically speaking relies on a
"lagrangian formulation" of the problem, albeit in a highly irregular setting.
We also establish stability of the solution in weak and strong topologies, and
propagation of the $BV$ regularity. In the case of either nonnegative or
nonpositive velocity fields we also establish a $BV$-in-time regularity result,
and we exhibit a counterexample showing that the result is false in the case of
sign-changing vector fields. To conclude, we establish a trace renormalization
property.
