Nonexistence of global characteristics for viscosity solutions

Roos, Valentine

doi:10.2140/apde.2020.13.1145

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Nonexistence of global characteristics for viscosity solutions

Valentine Roos

Vol. 13 (2020), No. 4, 1145–1172

DOI: 10.2140/apde.2020.13.1145

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. We prove that there exists no other class of integrable Hamiltonians sharing this property. To do so, we build for any nonconvex, nonconcave 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.

Keywords

Hamilton–Jacobi equation, nonconvex Hamiltonian dynamics, viscosity solution, variational solution, wavefronts, characteristics

Mathematical Subject Classification 2010

Primary: 49L25

Milestones

Received: 22 August 2018

Revised: 23 March 2019

Accepted: 18 April 2019

Published: 13 June 2020

Authors

Valentine Roos
UMPA ENS Lyon Lyon France

Vol. 13, No. 4, 2020