Particle dynamics inside shocks in Hamilton-Jacobi equations

Details Particle dynamics inside shocks in Hamilton-Jacobi equations Particle dynamics inside shocks in Hamilton-Jacobi equations

2010-01-04

arxiv.org/abs/1001.0498v1

Phil. Trans. R. Soc. A vol. 368 no. 1916 (2010) 1579-1593

Physics

Mathematical Physics

15 pages, no figures; to appear in Philos. Trans. R. Soc. series A

Scientific paper

10.1098/rsta.2009.0283

Characteristics of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. For nonsmooth "viscosity" solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that for any convex Hamiltonian there exists a uniquely defined canonical global nonsmooth coalescing flow that extends particle trajectories and determines dynamics inside the shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss relation to the "dissipative anomaly" in the limit of vanishing viscosity.

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