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Nonexistence of global characteristics for viscosity solutions

Abstract : 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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https://hal.archives-ouvertes.fr/hal-01827656
Contributor : Valentine Roos Connect in order to contact the contributor
Submitted on : Monday, July 2, 2018 - 3:27:26 PM
Last modification on : Friday, September 2, 2022 - 4:52:30 PM
Long-term archiving on: : Monday, October 1, 2018 - 7:46:11 AM

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  • HAL Id : hal-01827656, version 1
  • ARXIV : 1807.01038

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Valentine Roos. Nonexistence of global characteristics for viscosity solutions. Analysis & PDE, 2020. ⟨hal-01827656⟩

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