Integrability of the derivative of solutions to a singular one-dimensional parabolic problem

TL;DR

This paper investigates the integrability properties of derivatives of solutions to a singular one-dimensional parabolic PDE with periodic boundary conditions, focusing on gradient flows of convex functionals and their elliptic counterparts.

Contribution

It establishes integrability results for derivatives of solutions to a specific class of singular parabolic equations and their elliptic semidiscretizations, expanding understanding of solution regularity.

Findings

Proves integrability of derivatives for solutions with initial data in W^{1,1}

Extends results to elliptic semidiscretization problems

Focuses on gradient flows of convex functionals with linear growth

Abstract

We study integrability of the derivative of solutions to a singular one-dimensional parabolic equation with initial data in $W^{1,1}$. In order to avoid additional difficulties we consider only the periodic boundary conditions. The problem we study is a gradient flow of a convex, linear growth variational functional. We also prove a similar result for the elliptic companion problem, i.e. the time semidiscretization.