Particle dynamics inside shocks in Hamilton-Jacobi equations
The 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...
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| Veröffentlicht in: | Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences physical, and engineering sciences, 2010-04, Vol.368 (1916), p.1579-1593 |
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| container_issue | 1916 |
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| container_title | Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences |
| container_volume | 368 |
| creator | Khanin, Konstantin Sobolevski, Andrei |
| description | The 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 the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the 'dissipative anomaly' in the limit of vanishing viscosity. |
| doi_str_mv | 10.1098/rsta.2009.0283 |
| format | Article |
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| subjects | Burger equation Control Theory Convex Sets And Geometric Inequalities Flow velocity Hamilton Jacobi equation Hamiltonian Formulations Lagrangian function Mathematics Momentum Shock Wave Interactions And Shock Effects Singularity Theory Trajectories Uniqueness Velocity Viscosity |
| title | Particle dynamics inside shocks in Hamilton-Jacobi equations |
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