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Working paper

On dynamics of Lagrangian trajectories for Hamilton–Jacobi equations

arxiv.org. math. Cornell University, 2012. No. arXiv:1211.7084.
Khanin K., Sobolevski A.
Characteristic curves of a Hamilton–Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth “viscosity” solutions, which give rise to discontinuous velocity fields, this picture holds 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, a viscous regularization allows to construct a nonsmooth flow that extends particle tra- jectories and determines dynamics inside the shock manifolds. This flow consists of integral curves of a particular velocity field, which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds.