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Non-existence of global characteristics for viscosity solutions

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VarViscDiff.pdf (409.3Kb)
Date
2018
Publishing date
2018
Collection title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Link to item file
https://hal.archives-ouvertes.fr/hal-01827656
Dewey
Analyse
Sujet
viscosity solutions; Hamilton–Jacobi equation
URI
https://basepub.dauphine.fr/handle/123456789/17960
Collections
  • CEREMADE : Publications
Metadata
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Author
Roos, Valentine
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Document de travail / Working paper
Item number of pages
26
Abstract (EN)
Two different types of generalized solutions, namely viscosity and variational solutions, were introduced to solve the first-order evolutionary Hamilton–Jacobi equation. They coincide if the Hamiltonian is convex in the momentum variable. In this paper we prove that there exists no other class of integrable Hamiltonians sharing this property. To do so, we build for any non-convex non-concave integrable Hamiltonian a smooth initial condition such that the graph of the viscosity solution is not contained in the wavefront associated with the Cauchy problem. The construction is based on a new example for a saddle Hamiltonian and a precise analysis of the one-dimensional case, coupled with reduction and approximation arguments.

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