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

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

2003-12-20

arxiv.org/abs/math/0312395v2

"Idempotent Mathematics and Mathematical Physics", G. L. Litvinov, V. P. Maslov (eds.), AMS, Providence, 2005, ISBN 0-8218-353

Mathematics

Optimization and Control

19 pages, no figures; based on a talk given at the workshop "Idempotent Mathematics and Mathematical Physics" at the E. Schroe

Scientific paper

We 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.

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