Vol. 13, No. 4, 2020

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Nonexistence of global characteristics for viscosity solutions

Valentine Roos

Vol. 13 (2020), No. 4, 1145–1172

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.

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Hamilton–Jacobi equation, nonconvex Hamiltonian dynamics, viscosity solution, variational solution, wavefronts, characteristics
Mathematical Subject Classification 2010
Primary: 49L25
Received: 22 August 2018
Revised: 23 March 2019
Accepted: 18 April 2019
Published: 13 June 2020
Valentine Roos
ENS Lyon