Singularities of solutions of Hamilton-Jacobi equations

TL;DR

This survey reviews the progress in understanding how singularities in viscosity solutions of Hamilton-Jacobi equations propagate and their applications across geometry, dynamical systems, and PDEs over recent decades.

Contribution

It provides a comprehensive overview of the quantitative analysis of singularity propagation and highlights diverse applications of the theory.

Findings

Enhanced understanding of singularity behavior in viscosity solutions

Connections established between Hamilton-Jacobi theory and geometry

Applications demonstrated in dynamical systems and PDEs

Abstract

This is a survey paper on the quantitative analysis of the propagation of singularities for the viscosity solutions to Hamilton-Jacobi equations in the past decades. We also review further applications of the theory to various fields such as Riemannian geometry, Hamiltonian dynamical systems and partial differential equations.