Dynamic and asymptotic behavior of singularities of certain weak KAM solutions on the torus

TL;DR

This paper investigates the long-term behavior of generalized characteristics in Hamilton-Jacobi equations on the torus, linking their omega-limit sets to the projected Aubry set, thus advancing understanding of weak KAM solutions.

Contribution

It establishes a connection between the omega-limit sets of generalized characteristics semiflows and the projected Aubry set for certain Hamilton-Jacobi equations on the torus.

Findings

Characterization of omega-limit sets in terms of the projected Aubry set

Analysis of the asymptotic behavior of weak KAM solutions

Insights into the dynamical properties of generalized characteristics

Abstract

For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristics semiflows associated with certain Hamilton-Jacobi equations, and build the relation between the $\omega$-limit set of this semiflow and the projected Aubry set.