Total Variation Flow and Sign Fast Diffusion in one dimension

TL;DR

This paper analyzes the one-dimensional Total Variation Flow and Sign Fast Diffusion Equation, providing explicit dynamics and sharp asymptotic behavior, and establishing a direct correspondence between the two models.

Contribution

It offers explicit solutions and asymptotic analysis for TVF and SFDE in one dimension, revealing their connection and detailed behavior.

Findings

Explicit dynamic solutions for TVF

Sharp asymptotic behavior characterized

Established correspondence between TVF and SFDE

Abstract

We consider the dynamics of the Total Variation Flow (TVF) $u_t=\div(Du/|Du|)$ and of the Sign Fast Diffusion Equation (SFDE) $u_t=\Delta\sign(u)$ in one spatial dimension. We find the explicit dynamic and sharp asymptotic behaviour for the TVF, and we deduce the one for the SFDE by an explicit correspondence between the two equations.