Particle Dynamics Inside Shocks in HamiltonJacobi Equations
Abstract
Characteristic curves of a HamiltonJacobi 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.
 Publication:

Numerical Analysis and Applied Mathematics ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics
 Pub Date:
 September 2011
 DOI:
 10.1063/1.3636824
 Bibcode:
 2011AIPC.1389..690K
 Keywords:

 viscosity of liquids;
 velocity measurement;
 laminar flow;
 momentum;
 47.20.Gv;
 06.30.Gv;
 47.15.K;
 45.20.df;
 Viscous and viscoelastic instabilities;
 Velocity acceleration and rotation;
 Inviscid laminar flows;
 Momentum conservation