On the gradient flow of a one-homogeneous functional

TL;DR

This paper links the gradient flow of a one-homogeneous functional to a generalized Hele-Shaw flow, providing new variational insights, existence, and uniqueness results, along with explicit solutions and qualitative properties in one dimension.

Contribution

It establishes a novel connection between gradient flows of one-homogeneous functionals and Hele-Shaw flows, including variational representations and explicit solutions.

Findings

Proves the equivalence between gradient flow and Hele-Shaw flow formulations.

Provides existence and uniqueness results for weak solutions.

Derives explicit solutions for Total Variation flow in one dimension.

Abstract

We consider the gradient flow of a one-homogeneous functional, whose dual involves the derivative of a constrained scalar function. We show in this case that the gradient flow is related to a weak, generalized formulation of the Hele-Shaw flow. The equivalence follows from a variational representation, which is a variant of well-known variational representations for the Hele-Shaw problem. As a consequence we get existence and uniqueness of a weak solution to the Hele-Shaw flow. We also obtain an explicit representation for the Total Variation flow in one dimension and easily deduce basic qualitative properties, concerning in particular the "staircasing effect".