Konstantin Khanin, Andrei Sobolevski
Characteristic curves of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. This description, however, is valid only for smooth solutions. For non-smooth "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 non-smooth flow that extends particle trajectories 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.
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http://arxiv.org/abs/1211.7084
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