Energy solutions to one-dimensional singular parabolic problems with $BV$ data are viscosity solutions

TL;DR

This paper investigates one-dimensional singular parabolic equations with BV initial data, establishing the existence of solutions in the energy space and proving they are viscosity solutions in the Giga-Giga sense.

Contribution

It demonstrates the existence of solutions in the BV energy space and proves their equivalence to viscosity solutions for a class of singular parabolic problems.

Findings

Existence of solutions in BV energy space.

Solutions are viscosity solutions in the Giga-Giga sense.

Applicable to one-dimensional singular parabolic equations.

Abstract

We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in $BV$, which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.