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.
Highlights
This is a survey paper concerning the progress made for the singularities of the solutions to Hamilton–Jacobi equations in the past decades
We review further applications of the theory to various fields such as Riemannian geometry, Hamiltonian dynamical systems and partial differential equations
Hamilton–Jacobi equation, viscosity solution, propagation of singularities, singular characteristics
Summary
This is a survey paper concerning the progress made for the singularities of the solutions to Hamilton–Jacobi equations in the past decades. The propagation of singularities of semiconcave functions along Lipschitz arcs was firstly studied in [3] and extend to solutions of Hamilton–Jacobi equations [5]. It is the first time in [5] the authors introduced the important notion of generalized characteristics for Hamilton–Jacobi equation (HJ), which is a keystone for the further progress later. Inspired by earlier works [19,20,77], Khanin and Sobolevski essentially proved the existence of singular characteristics satisfying (1.1) without convex hull under some extra conditions on the initial data [63].
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