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

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 ω-limit set of this semiflow and the projected Aubry set.

Singularities of solutions of Hamilton-Jacobi equations

“…However, the uniqueness issue is still open for higher dimensional manifolds. Recalling some results in [CCC19], assuming uniqueness for generalized characteristics, one can bridge the Aubry set (Mather set) and the invariant set of the associated semi-flow of generalized characteristics. Recently, relations between propagation of singularities and global dynamics of lower dimensional Hamiltonian systems have also been pointed out in [Zha20].…”

On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem

“…It is not clear whether Lemma 2.6 holds true without the assumption that φ is bounded in general, while it does for the Lagrangian in the form L(x, u, v) = −λu + L 0 (x, v), λ > 0, which is the Lagrangian with respect to the well known discounted Hamiltonian (see, for instance, [12]). Lemma 2.6 not only ensures that the infimum in (2.2) is indeed minimum if φ is a bounded and Lipschitz continuous function on R n , but also plays an essential part for the applications to the study of the global propagation of singularities of the associated Hamilton-Jacobi equations ( [3], [4], [8] and [2]). Lemma 2.8.…”

Herglotz' variational principle and Lax-Oleinik evolution

“…then the Lipschitz and semiconcavity estimates can be obtained directly from the uniform Lipschitz and semiconcavity estimates for h L (see, for instance, [19,Theorem 3.4.4]). This is also a key point of our program for the study of the propagation of singularities of viscosity solutions (see, for instance, [13,16,14,12]).…”