Multi-time systems of conservation laws
Authors:
Aldo Bazan, Paola Loreti and Wladimir Neves
Journal:
Quart. Appl. Math. 72 (2014), 491-511
MSC (2010):
Primary 54C40, 14E20; Secondary 46E25, 20C20
DOI:
https://doi.org/10.1090/S0033-569X-2014-01341-3
Published electronically:
June 25, 2014
MathSciNet review:
3237561
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: Motivated by the work of P. L. Lions and J.-C. Rochet (1986) concerning multi-time Hamilton-Jacobi equations, we introduce the theory of multi-time systems of conservation laws. We show the existence and uniqueness of solution to the Cauchy problem for a system of multi-time conservation laws with two independent time variables in one space dimension. Our proof relies on a suitable generalization of the Lax-Oleinik formula.
References
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- J.-C. Rochet, The taxation principle and multitime Hamilton-Jacobi equations, J. Math. Econom. 14 (1985), no. 2, 113–128. MR 827147, DOI https://doi.org/10.1016/0304-4068%2885%2990015-1
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References
- O. Alvarez, E. N. Barron, and H. Ishii, Hopf-Lax formulas for semicontinuous data, Indiana Univ. Math. J. 48 (1999), no. 3, 993–1035. MR 1736971 (2001c:35048), DOI https://doi.org/10.1512/iumj.1999.48.1648
- M. Bardi and L. C. Evans, On Hopf’s formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal. 8 (1984), no. 11, 1373–1381. MR 764917 (85k:35043), DOI https://doi.org/10.1016/0362-546X%2884%2990020-8
- Guy Barles and Agnès Tourin, Commutation properties of semigroups for first-order Hamilton-Jacobi equations and application to multi-time equations, Indiana Univ. Math. J. 50 (2001), no. 4, 1523–1544. MR 1889069 (2003a:49037), DOI https://doi.org/10.1512/iumj.2001.50.1925
- Franco Cardin and Claude Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J. 144 (2008), no. 2, 235–284. MR 2437680 (2010a:37107), DOI https://doi.org/10.1215/00127094-2008-036
- M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), no. 2, 487–502. MR 732102 (86a:35031), DOI https://doi.org/10.2307/1999247
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001)
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845 (99e:35001)
- Gu, G. Q., Chung, K. H., Hui, P. M., Two-dimensional traffic flow problems in inhomogeneous lattices, Physica A, 217 (1995), 339–347.
- Steven S. Gubser, The little book of string theory, Science Essentials, Princeton University Press, Princeton, NJ, 2010. MR 2655897 (2011e:81215)
- Cyril Imbert and Michel Volle, On vectorial Hamilton-Jacobi equations: Well-posedness in optimization and related topics (Warsaw, 2001), Control Cybernet. 31 (2002), no. 3, 493–506. MR 1978737 (2004d:35036)
- Kruzkov S., First-order quasilinear equations with several space variables, Math. USSR Sb. 10 (1970), 217–243.
- P.-L. Lions and J.-C. Rochet, Hopf formula and multitime Hamilton-Jacobi equations, Proc. Amer. Math. Soc. 96 (1986), no. 1, 79–84. MR 813815 (87h:35056), DOI https://doi.org/10.2307/2045657
- Moreau, J. J., Fonctionnelles convexes, Séminaire Jean Leray, no. 2 (1966-1967), 1–108.
- Monica Motta and Franco Rampazzo, Nonsmooth multi-time Hamilton-Jacobi systems, Indiana Univ. Math. J. 55 (2006), no. 5, 1573–1614. MR 2270930 (2008a:35025), DOI https://doi.org/10.1512/iumj.2006.55.2760
- Mircea Neagu and Constantin Udrişte, A Riemann-Lagrange geometrization for metrical multi-time Lagrange spaces, Balkan J. Geom. Appl. 11 (2006), no. 1, 87–98. MR 2230663 (2007b:53041)
- Sławomir Plaskacz and Marc Quincampoix, Oleinik-Lax formulas and multitime Hamilton-Jacobi systems, Nonlinear Anal. 51 (2002), no. 6, 957–967. MR 1926078 (2004i:49058), DOI https://doi.org/10.1016/S0362-546X%2801%2900871-9
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683 (43 \#445)
- J.-C. Rochet, The taxation principle and multitime Hamilton-Jacobi equations, J. Math. Econom. 14 (1985), no. 2, 113–128. MR 827147 (87g:90039), DOI https://doi.org/10.1016/0304-4068%2885%2990015-1
- J. Stickforth, The Kepler problem in relativistic multi-time formalism. I, II, Acta Mech. 123 (1997), no. 1-4, 187–193, 195–201. MR 1474558 (98m:70026), DOI https://doi.org/10.1007/BF01178409
- A. I. Vol′pert, Spaces $\textrm {BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338 (35 \#7172)
- M. Zavidovique, Weak KAM for commuting Hamiltonians, Nonlinearity 23 (2010), no. 4, 793–808. MR 2602014 (2011g:37177), DOI https://doi.org/10.1088/0951-7715/23/4/002
- Barton Zwiebach, A first course in string theory, 2nd ed., Cambridge University Press, Cambridge, 2009. With a foreword by David Gross. MR 2477975 (2011f:81179)
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Additional Information
Aldo Bazan
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária 21945-970, Rio de Janeiro, Brazil
Email:
aabp2003@pg.im.ufrj.br
Paola Loreti
Affiliation:
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, via A. Scarpa n.16 00161 Roma
Email:
paolaloreti@gmail.com
Wladimir Neves
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária 21945-970, Rio de Janeiro, Brazil
Email:
wladimir@im.ufrj.br
Keywords:
Conservation laws,
Hamilton-Jacobi equations,
Cauchy problem,
multi-time partial differential equations.
Received by editor(s):
June 29, 2012
Published electronically:
June 25, 2014
Additional Notes:
Aldo Bazan is supported by FAPERJ by the grant 2009.2848.0.
Wladimir Neves is partially supported by FAPERJ through the grant E-26/ 111.564/2008 entitled “Analysis, Geometry and Applications” and by Pronex-FAPERJ through the grant E-26/ 110.560/2010 entitled "Nonlinear Partial Differential Equations".
Article copyright:
© Copyright 2014
Brown University
