Global Propagation of Singularities for Time Dependent Hamilton-Jacobi Equations

TL;DR

This paper studies how singularities in solutions to time-dependent Hamilton-Jacobi equations evolve, showing they cannot disappear in finite time and analyzing energy behavior along special characteristic curves.

Contribution

It introduces estimates on energy dissipation along generalized characteristics and proves the non-vanishing of singularities over time.

Findings

Singularities propagate along generalized characteristics.

Energy along special curves dissipates in a controlled manner.

Singularities cannot vanish in finite time for viscosity solutions.

Abstract

We investigate the properties of the set of singularities of semiconcave solutions of Hamilton-Jacobi equations of the form \begin{equation*} u_t(t,x)+H(\nabla u(t,x))=0, \qquad\text{a.e. }(t,x)\in (0,+\infty)\times\Omega\subset\mathbb{R}^{n+1}\,. \end{equation*} It is well known that the singularities of such solutions propagate locally along generalized characteristics. Special generalized characteristics, satisfying an energy condition, can be constructed, under some assumptions on the structure of the Hamiltonian $H$. In this paper, we provide estimates of the dissipative behavior of the energy along such curves. As an application, we prove that the singularities of any viscosity solution of the above equation cannot vanish in a finite time.