Particle Dynamics Inside Shocks in Hamilton-Jacobi Equations
Abstract
Characteristic curves of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. For non-smooth "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 non-smooth 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