On Dynamics of Lagrangian Trajectories for Hamilton–Jacobi Equations

Abstract

Characteristic curves of a Hamilton–Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth “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 discuss two physically meaningful regularization procedures, one corresponding to vanishing viscosity and another to weak noise limit. We show that for any convex Hamiltonian, a viscous regularization allows us to construct a nonsmooth flow that extends particle trajectories and determines dynamics inside the shock manifolds. This flow consists of integral curves of a particular “effective” velocity field, which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds. The effective velocity field arising in the weak noise limit is generally non-unique and different from the viscous one, but in both cases there is a fundamental self-consistency condition constraining the dynamics.

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Correspondence to Andrei Sobolevski.

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We acknowledge the support of the French Ministry for Science and Higher Education. A.S. gratefully acknowledges the support of the Simons-IUM fellowship and hospitality of Department of Mathematics, University of Toronto. K.K. acknowledges support of the NSERC Discover Grant.

Communicated by C. Dafermos

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Khanin, K., Sobolevski, A. On Dynamics of Lagrangian Trajectories for Hamilton–Jacobi Equations. Arch Rational Mech Anal 219, 861–885 (2016). https://doi.org/10.1007/s00205-015-0910-x

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Keywords

  • Viscosity Solution
  • Integral Curve
  • Burger Equation
  • Jacobi Equation
  • Integral Curf