Global Propagation of Singularities for Time Dependent Hamilton-Jacobi\n Equations

2014/08/24 by Piermarco Cannarsa, Cannarsa, Piermarco, Marco Mazzola +3 · 2 citations
Mathematics · Physics and Astronomy · #53C35 #58F15 #58F17 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1408.5613

openalex publication_date 2014/08/24 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We investigate the properties of the set of singularities of semiconcave\nsolutions of Hamilton-Jacobi equations of the form n u_t(t,x)+H(
nabla u(t,x))=0,
qquad
texta.e. (t,x)
in\n(0,+
infty)
times
Omega
subset
mathbbRⁿ⁺¹
,. It is well\nknown that the singularities of such solutions propagate locally along\ngeneralized characteristics. Special generalized characteristics, satisfying an\nenergy condition, can be constructed, under some assumptions on the structure\nof the Hamiltonian H. In this paper, we provide estimates of the dissipative\nbehavior of the energy along such curves. As an application, we prove that the\nsingularities of any viscosity solution of the above equation cannot vanish in\na finite time.\n

Citations

• The continuous dependence of generalized solutions of non‐linear partial differential equations upon initial data

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