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

TL;DR

This paper proves BV regularity for solutions to certain non-linear parabolic PDEs with linear drift, covering boundary conditions and general nonlinearities, extending to linear transport equations.

Contribution

It establishes BV regularity results for non-linear parabolic PDEs with linear drift in bounded domains under broad conditions.

Findings

BV regularity of solutions proven

Results include linear transport equations as special case

Applicable to mixed boundary conditions and general nonlinearities

Abstract

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.