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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Bec, J., Khanin, K.: Burgers turbulence. Phys. Rep. 447(1–2), 1–66, 2007
Bogaevsky I.A.: Perestroikas of shocks and singularities of minimum functions. Phys. D: Nonlinear Phenom. 173(1–2), 1–28 (2002)
Bogaevsky, I.A.: Matter Evolution in Burgulence. arXiv:math-ph/0407073, 2004
Bogaevsky I.A.: Discontinuous gradient differential equations and trajectories in the calculus of variations. Sbornik Math. 197(12), 1723–1751 (2006)
Bregman L.M.: A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. and Math. Phys. 7, 200–217 (1967)
Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematical Studies, vol. 5. North-Holland, Amsterdam, 1973
Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser Boston, Inc., Boston, MA, 2004
Cannarsa, P., Yu, Y.: Singular dynamics for semiconcave functions. J. Eur. Math. Soc. 2009
Clarke, F.H.: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, vol. 5. SIAM, 1990
Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67, 1992
Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, 2 edn. springer, 2005
Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics. Cambridge Univ. Press, Cambridge, 2016
Fleming, W.H., Soner, H.M.: Controlled Markov processes and viscosity solutions, Stochastic Modelling and Applied Probability, vol. 25, 2 edn. Springer, New York, 2005
Gangbo W., McCann R.J.: The geometry of optimal transportation. Acta Math. 177(2), 113–161 (1996)
Hopf, E.: The partial differential equation u t + uu x = μu xx . Commun. Pure Appl. Math. 3, 201–230, 1950
Khanin K., Sobolevski A.: Particle dynamics inside shocks in hamilton-jacobi equations. Philos. Trans. R. Soc. Lond. 368, 1579–1593 (2010)
Khesin, B.A., Misiołek, G.: Analysis and Singularities, Tr. Mat. Inst. Steklova, vol. 259, chap. Shock waves for the Burgers equation and curvatures of diffeomorphismgroups. Nauka, Moscow, 2007
Kružkov, S.N.: Generalized solutions of hamilton–jacobi equations of eikonal type. Mat. Sb. 98(140)(3(11)), 450–493, 1975
Lax P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computaton. Commun. Pure Appl. Math. 7, 159–193 (1954)
Lax P.D.: Hyperbolic systems of conservation laws. cpam 10, 537–566 (1957)
Lions, P.L.: Generalized solutions of Hamilton–Jacobi equations, Research Notes in Mathematics, vol. 69. Pitman (Advanced Publishing Program), Boston, Mass.–London, 1982
Olenik O.A.: On cauchy’s problem for nonlinear equations in a class of discontinuous functions. Doklady Akad. Nauk SSSR (N.S.) 95(3), 451–455 (1954)
Rockafellar, R.T.: Convex analysis, Princeton Mathematical Series, vol. 28. Princeton Univ. Press, Princeton, NJ, 1970
Strömberg T.: Propagation of singularities along broken characteristics. Nonlinear Anal.: Theory Methods Appl. 85, 93–109 (2013)
Villani, C.: Optimal transport: Old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. springer, 2009
Weinan, E., Khanin, K., Mazel, A., Sinai, Y.: Invariant measures for burgers equation with stochastic forcing. Ann. Math. (2) 151(3), 877–960, 2000
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
About this article
Cite this article
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
- Viscosity Solution
- Integral Curve
- Burger Equation
- Jacobi Equation
- Integral Curf