A blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain

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dc.creatorKhanin, Konstantin
dc.creatorKhmelev, Dmitry
dc.creatorSobolevskii, Andrei
dc.date2003-12-20
dc.date2005-04-07
dc.date.accessioned2026-07-07T05:04:06Z
dc.date.available2026-07-07T05:04:06Z
dc.descriptionWe construct an example of blow-up in a flow of min-plus linear operators arising as solution operators for a Hamilton-Jacobi equation with a Hamiltonian of the form |p|^alpha+U(x,t), where alpha>1 and the potential U(x,t) is uniformly bounded together with its gradient. The construction is based on the fact that, for a suitable potential defined on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O((log T)^(2-2/alpha)). We also show that this growth estimate cannot be surpassed. Implications of this example for existence of global generalized solutions to randomly forced Hamilton-Jacobi or Burgers equations are discussed.
dc.description19 pages, no figures; based on a talk given at the workshop "Idempotent Mathematics and Mathematical Physics" at the E. Schroedinger Institute for Mathematical Physics in Vienna in February 2003. A dedication indicating the untimely death of Dmitry Khmelev is added
dc.identifierhttps://arxiv.org/abs/math/0312395
dc.identifierhttp://arxiv.org/abs/math/0312395
dc.identifier"Idempotent Mathematics and Mathematical Physics", G. L. Litvinov, V. P. Maslov (eds.), AMS, Providence, 2005, ISBN 0-8218-3538-6, p. 161-179
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/69670
dc.subjectOptimization and Control
dc.subjectDynamical Systems
dc.subject35L67; 49L99
dc.titleA blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain
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