Calculus of Variations and Geometric Measure Theory

S. Dovetta - E. Marconi - L. V. Spinolo

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

created by marconi on 24 May 2021
modified on 29 Oct 2021


Accepted Paper

Inserted: 24 may 2021
Last Updated: 29 oct 2021

Journal: Journal of Differential Equations
Year: 2021


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.