Abstract
We investigate the vanishing viscosity limit for Hamilton–Jacobi PDE with nonconvex Hamiltonians, and present a new method to augment the standard viscosity solution approach. The main idea is to introduce a solution σ ε of the adjoint of the formal linearization, and then to integrate by parts with respect to the density σ ε. This procedure leads to a natural phase space kinetic formulation and also to a new compensated compactness technique.
Article PDF
Similar content being viewed by others
References
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi-Bellman Equations, with appendices by M. Falcone and P. Soravia, Birkhä user, 1997
Ballou D.: Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions. Trans. Am. Math. Soc. 152, 441–460 (1970)
Bagnerini P., Rascle M., Tadmor E.: Compensated compactness for 2D conservation laws. J. Hyperbolic Differ. Equ. 2, 697–712 (2005)
Christ, M.: On linear functionals which are positive on convex functions (preprint)
Crandall M.G., Evans L.C., Lions P.-L.: Some properties of viscosity solutions of Hamilton Jacobi equations. Trans. Am. Math. Soc. 282, 487–502 (1984)
Crandall M.G., Lions P.-L.: Two approximations of solutions of Hamilton Jacobi equations. Math. Comp. 43, 1–19 (1984)
Dafermos C.: Regularity and large time behavior of solutions of a conservation law without convexity. Proc. R. Soc. Edinb. Sect. A 99, 201–239 (1985)
Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 74, American Mathematical Society, 1990
Evans L.C.: A survey of partial differential equations methods in weak KAM theory. Comm. Pure Appl. Math. 57, 445–480 (2004)
Federer, H.: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer, 1969
Fleming W.: The convergence problem for differential games II. Advances in Game Theory, Princeton University Press, Princeton, 195–210, 1964
Fleming W.: The Cauchy problem for degenerate parabolic equations. J. Math. Mech. 13, 987–1008 (1964)
Glimm J., Kranzer H.C., Tan D., Tangerman F.M.: Wave fronts for Hamilton–Jacobi equations: the general theory for Riemann solutions in \({\mathbb R^n}\). Comm. Math. Phys. 187, 647–677 (1997)
Marson A.: Nonconvex conservation laws and ordinary differential equations. J. London Math. Soc. 2(69), 428–440 (2004)
Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations, translated by S. K. Mitter, Springer, 1971
Lions P.-L.: Identification du cone dual des fonctions convexes et applications. C. R. Acad. Sci. Paris Sr. I Math. 326, 1385–1390 (1998)
Lions P.-L., Perthame B., Tadmor E.: Kinetic formulation of the isentropic gas dynamics and p-systems. Comm. Math. Phys. 163, 415–431 (1994)
Lions P.-L., Perthame B., Tadmor E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7, 169–191 (1994)
Perthame, B.: Kinetic Formulation of Conservation Laws. Oxford Lecture Series in Mathematics and its Applications, vol. 21. Oxford University Press, Oxford, 2002
Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Vol. IV, Research Notes in Math 39, Pitman, 136–212, 1979
Tran, H.V.: Paper to appear
Zheng, Y.: Systems of Conservation Laws. Two-dimensional Riemann problems. Progress in Nonlinear Differential Equations and their Applications, 38. Birkhäuser, 2001
Acknowledgements
I thank E. Barron, Y. Brenier, M. Christ, C. Dafermos, R. Jensen and H. Zhao for providing me with references and suggestions. I am particularly grateful to I. Strub for explaining to me uses of the adjoint method for the optimal control of PDE. I also thank the referee for carefully reading this paper and making many useful suggestions.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Dafermos
L. C. Evans was supported in part by NSF Grant DMS-0500452.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Evans, L.C. Adjoint and Compensated Compactness Methods for Hamilton–Jacobi PDE. Arch Rational Mech Anal 197, 1053–1088 (2010). https://doi.org/10.1007/s00205-010-0307-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-010-0307-9