$BV$-Estimates for Non-Linear Parabolic PDE with Linear Drift

In the present work, we establish space Bounded Variation $(BV)$ regularity of the solution for a non-linear parabolic partial differential equations involving a linear drift term. We study the problem in a bounded domain with mixed Dirichlet-Neumann boundary conditions, a general non-linearity and reasonable assumptions on the data. Our results also cover, as a particular case, the linear transport equation in a bounded domain with an outward-pointing drift vector field on the boundary.