Long time behaviour of generalised gradient flows via occupational measures

TL;DR

This paper studies the long-term behavior of generalized gradient flows for Hamiltonian systems on tori, using occupational measures to determine whether solutions approach regular or singular critical sets over time.

Contribution

It introduces a novel approach using limiting occupational measures to analyze the asymptotic behavior of generalized gradient flows on flat tori.

Findings

Occupational measures help identify the flow's approach to critical sets.

The flow can reach the singular set in finite time or approach the regular set asymptotically.

Method provides criteria for long-term behavior of solutions.

Abstract

This paper introduces new methods to study the long time behaviour of the generalised gradient flow associated with a solution of the critical equation for mechanical Hamiltonian system posed on the flat torus $\mathbb{T}^d$. For this analysis it is necessary to look at the critical set of $u$ consisting of all the points on $\mathbb{T}^d$ such that zero belongs to the super-differential of such a solution. Indeed, such a set turns out to be an attractor for the generalised gradient flow. Moreover, being the critical set the union of two subsets of rather different nature, namely the regular critical set and the singular set, we are interested in establishing whether the generalised gradient flow approaches the former or the latter as $t\to \infty$. One crucial tool of our analysis is provided by limiting occupational measures, a family of measures that are invariant under the generalized flow. Indeed, we show that by integrating the potential with respect to such measures, one can deduce whether the generalised gradient flow enters the singular set in finite time, or it approaches the regular critical set as time tends to infinity.