In the context of weak KAM theory, we discuss the commutators $\{T^-_t\circ T^+_t\}_{t\geqslant0}$ and $\{T^+_t\circ T^-_t\}_{t\geqslant0}$ of Lax-Oleinik operators. We characterize the relation $T^-_t\circ T^+_t=Id$ for both small time and arbitrary time $t$. We show this relation characterizes controllability for evolutionary Hamilton-Jacobi equation. Based on our previous work on the cut locus of viscosity solution, we refine our analysis of the cut time function $\tau$ in terms of commutators $T^+_t\circ T^-_t-T^+_t\circ T^-_t$ and clarify the structure of the super/sub-level set of the cut time function $\tau$.

This is our first paper on the extension of our recent work on the Lax-Oleinik commutators and its applications to the intrinsic approach of propagation of singularities of the viscosity solutions of Hamilton-Jacobi equations. We reformulate Kantorovich-Rubinstein duality theorem in the theory of optimal transport in terms of abstract Lax-Oleinik operators, and analyze the relevant optimal transport problem in the case the cost function c(x,y)=h(t_1,t_2,x,y) is the fundamental solution of Hamilton-Jacobi equation. For further applications to the problem of cut locus and propagation of singularities in optimal transport, we introduce corresponding random Lax-Oleinik operators. We also study the problem of singularities for c-concave functions and its dynamical implication when c is the fundamental solution with t_2-t_1≪1 and t_2-t_1<∞, and c is the Peierls' barrier respectively.

The notion of strict singular characteristics is important in the wellposedness issue of singular dynamics on the cut locus of the viscosity solutions. We provide an intuitive and rigorous proof of the existence of the strict singular characteristics of Hamilton–Jacobi equation H(x,Du(x),u(x))=0 in two dimensional case. We also proved if x is a strict singular characteristic, then we really have the right-differentiability of x and the right-continuity of ẋ^+(t) for every t . Such a strict singular characteristic must give a selection p(t)∈ D^+u(x(t)) such that p(t)=min _p∈ D^+u(x(t))H(x(t),p,u(x(t))) .

Main purpose of this paper is to study the local propagation of singularities of viscosity solution to contact type evolutionary Hamilton-Jacobi equationDtu(t,x)+H(t,x,Dxu(t,x),u(t,x))=0. An important issue of this topic is the existence, uniqueness and regularity of the strict singular characteristic. We apply the recent existence and regularity results on the Herglotz' type variational problem to the aforementioned Hamilton-Jacobi equation with smooth initial data. We obtain some new results on the local structure of the cut set of the viscosity solution near non-conjugate singular points. Especially, we obtain an existence result of smooth strict singular characteristic from and to non-conjugate singular initial point based on the structure of the superdifferential of the solution, which is even new in the classical time-dependent case. We also get a global propagation result for the C1 singular support in the contact case.

The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation \begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document} with fixed constant \begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}. We reduce the problem for equation \begin{document}$(\mathrm{HJ}_{\lambda})$\end{document} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of \begin{document}$(\mathrm{HJ}_{\lambda})$\end{document} propagate along locally Lipschitz singular characteristics \begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document} and time \begin{document}$ t $\end{document} can extend to \begin{document}$ +\infty $\end{document}. Essentially, we use \begin{document}$ \sigma $\end{document}-compactness of the Euclidean space which is different from the original construction in [4]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of \begin{document}$ u $\end{document} and the complement of Aubry set using the basic idea from [9].

We discuss various kinds of representation formulas for the viscosity solutions of the contact type Hamilton-Jacobi equations by using the Herglotz’ variational principle.