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Particle dynamics inside shocks in Hamilton-Jacobi equations.

Konstantin Khanin1, Andrei Sobolevski

  • 1Department of Mathematics, University of Toronto, Toronto, Ontario, Canada. khanin@math.toronto.edu

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 10, 2010
PubMed
Summary

This study introduces a global non-smooth coalescing flow for Hamilton-Jacobi equations, extending particle trajectories through shocks. This flow uniquely defines dynamics within shocks, offering a new perspective on fluid particle motion and viscosity solutions.

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Area of Science:

  • Fluid dynamics
  • Partial differential equations
  • Hamilton-Jacobi theory

Background:

  • Characteristic curves in Hamilton-Jacobi equations represent action-minimizing fluid particle trajectories.
  • Non-smooth viscosity solutions lead to discontinuous velocity fields, complicating trajectory analysis at shocks.

Purpose of the Study:

  • To define a global, non-smooth coalescing flow for Hamilton-Jacobi equations.
  • To extend particle trajectories beyond shock formations.
  • To describe dynamics within shocks and their relation to dissipative anomalies.

Main Methods:

  • Analysis of Hamilton-Jacobi equations with convex Hamiltonians.
  • Development of a canonical, non-smooth coalescing flow.
  • Variational description of effective velocity fields within shocks.

Main Results:

  • Existence and uniqueness of a global non-smooth coalescing flow.
  • Extension of particle trajectories through shocks.
  • Variational characterization of internal shock dynamics.

Conclusions:

  • The proposed flow provides a complete description of particle trajectories, including shock interiors.
  • This framework offers insights into the 'dissipative anomaly' in vanishing viscosity limits.
  • Establishes a novel approach to understanding non-smooth solutions in fluid dynamics.