Local strict singular characteristics: Cauchy problem with smooth initial data

TL;DR

This paper investigates the local behavior and propagation of singularities in viscosity solutions to contact type Hamilton-Jacobi equations with smooth initial data, introducing new results on strict singular characteristics and the structure of the cut set.

Contribution

It provides new existence and regularity results for strict singular characteristics and the local structure of the cut set, especially near non-conjugate singular points, using recent variational methods.

Findings

Existence of smooth strict singular characteristics from non-conjugate singular points.

New insights into the local structure of the cut set near singular points.

Global propagation results for the $C^1$ singular support in the contact case.

Abstract

Main purpose of this paper is to study the local propagation of singularities of viscosity solution to contact type evolutionary Hamilton-Jacobi equation $$ D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0. $$ 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 $C^1$ singular support in the contact case.