Initial–boundary value problems for merely bounded nearly incompressible vector fields in one space dimension

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