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Hyperbolic Conservation Laws on Spacetimes

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Nonlinear Conservation Laws and Applications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 153))

Abstract

We present a generalization of Kruzkov’s theory to manifolds. Nonlinear hyperbolic conservation laws are posed on a differential (n + 1)-manifold, called a spacetime, and the flux field is defined as a field of n-forms depending on a parameter. The entropy inequalities take a particularly simple form as the exterior derivative of a family of n-form fields. Under a global hyperbolicity condition on the spacetime, which allows arbitrary topology for the spacelike hypersurfaces of the foliation, we establish the existence and uniqueness of an entropy solution to the initial value problem, and we derive a geometric version of the standard L 1 semi-group property. We also discuss an alternative framework in which the flux field consists of a parametrized family of vector fields.

AMS(MOS) subject classifications. Primary: 35L65. Secondary: 76L05, 76N.

The author was supported by the Agence Nationale de la Recherche via the grant ANR 06-2-134423.

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References

  1. Amorim P., Ben-Artzi M., and LeFloch P.G., Hyperbolic conservation laws on manifolds: total variation estimates and the finite volume method, Meth. Appl. Anal. 12 (2005), 291–324.

    MATH  MathSciNet  Google Scholar 

  2. Amorim P., LeFloch P.G., and Okutmustur B., Finite volume schemes on Lorentzian manifolds, Comm. Math. Sc. 6 (2008), 1059–1086.

    MATH  MathSciNet  Google Scholar 

  3. Amorim P., LeFloch P.G., and Neves W., A geometric approach to error estimates for conservation laws posed on a spacetime, preprint ArXiv:1002.3137.

    Google Scholar 

  4. Ben-Artzi M. and LeFloch P.G., The well-posedness theory for geometry compatible, hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré: Nonlin. Anal. 24 (2007), 989–1008.

    Article  MATH  MathSciNet  Google Scholar 

  5. Ben-Artzi M., Falcovitz J., and LeFloch P.G., Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme, J. Comput. Phys. 228 (2009), 5650–5668.

    Article  MATH  MathSciNet  Google Scholar 

  6. Cockburn B., Coquel F., and LeFloch P.G., Convergence of finite volume methods, I.M.A. Series # 771, Minneapolis, February 1991.

    Google Scholar 

  7. Cockburn B., Coquel F., and LeFloch P.G., Convergence of finite volume methods for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), 687–705.

    Article  MATH  MathSciNet  Google Scholar 

  8. Coquel F. and LeFloch P.G., Convergence of finite difference schemes for conservation laws in several space dimensions, C.R. Acad. Sci. Paris Ser. I 310 (1990), 455–460.

    MATH  MathSciNet  Google Scholar 

  9. Coquel F. and LeFloch P.G., Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory, SIAM J. Numer. Anal. 30 (1993), 675–700.

    Article  MATH  MathSciNet  Google Scholar 

  10. Coquel F. and LeFloch P.G., Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp. 57 (1991), 169–210.

    Article  MATH  MathSciNet  Google Scholar 

  11. DiPerna R.J., Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), 223–270.

    Article  MATH  MathSciNet  Google Scholar 

  12. Eymard R., Galloüet T., and Herbin R., The finite volume method, in “Hand-book of Numerical Analysis”, Vol. VII, North-Holland, Amsterdam, 2000, pp. 713–1020.

    Google Scholar 

  13. Giesselman J. and LeFloch P.G., Hyperbolic conservation laws on spacetimes with boundary, in preparation.

    Google Scholar 

  14. Kröner D., Finite volume schemes in multidimensions, in “Numerical analysis” 1997 (Dundee), Pitman Res. Notes Math. Ser. 380, Longman, Harlow, 1998, pp. 179–192.

    Google Scholar 

  15. Kröner D., Noelle S., and Rokyta M., Convergence of higher-order upwind finite volume schemes on unstructured grids for scalar conservation laws with several space dimensions, Numer. Math. 71 (1995), 527–560.

    Article  MATH  MathSciNet  Google Scholar 

  16. Kruzkov S.N., First-order quasilinear equations with several space variables, Math. USSR Sb. 10 (1970), 217–243.

    Article  Google Scholar 

  17. LeFloch P.G. and Nédélec J., Explicit formula for weighted scalar nonlinear conservation laws, Trans. Amer. Math. Soc. 308 (1988), 667–683.

    Article  MathSciNet  Google Scholar 

  18. LeFloch P.G., Neves W., and Okutmustur B., Hyperbolic conservation laws on manifolds. Error estimate for finite volume schemes, Acta Math. Sinica 25 (2009), 1041–1066.

    Article  MATH  MathSciNet  Google Scholar 

  19. LeFloch P.G. and Okutmustur B., Conservation laws on manifolds with limited regularity, C.R. Acad. Sc. Paris, Ser. I 346 (2008), 539–543.

    MATH  MathSciNet  Google Scholar 

  20. LeFloch P.G. and Okutmustur B., Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms, Far East J. Math. Sci. 31 (2008), 49–83. (See also arXiv:0810.0255.)

    MATH  MathSciNet  Google Scholar 

  21. Panov E.Y., On the Cauchy problem for a first-order quasilinear equation on a manifold, Differential Equations 33 (1997), 257–266.

    MATH  MathSciNet  Google Scholar 

  22. Panov E.Y., On the Dirichlet problem for first-order quasilinear equations on a manifold, Trans. Amer. Math. Soc., 2010, to appear.

    Google Scholar 

  23. Szepessy A., Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions, Math. Comp. 53 (1989), 527–545.

    Article  MATH  MathSciNet  Google Scholar 

  24. Szepessy A., Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions, RAIRO Modél. Math. Anal. Numér. 25 (1991), 749–782.

    MATH  MathSciNet  Google Scholar 

  25. Tadmor E., Approximate solutions of nonlinear conservation laws, in “Advanced numerical approximation of nonlinear hyperbolic equations”, Cetraro, 1997, Lect. Notes in Math., 1697, Springer Berlin, 1998, pp. 1–149.

    Chapter  Google Scholar 

  26. Tadmor E., Rascle M., and Bagnerini P., Compensated compactness for 2D conservation laws, J. Hyperbolic Differ. Equ. 2 (2005), 697–712.

    Article  MATH  MathSciNet  Google Scholar 

  27. Westdickenberg M. and Noelle S., A new convergence proof for finite volume schemes using the kinetic formulation of conservation laws, SIAM J. Numer. Anal. 37 (2000), 742–757.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252, Paris, France

    Philippe G. Lefloch

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  1. Philippe G. Lefloch
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Editors and Affiliations

  1. McAllister Bldg 201, University Park, 16802-6401, Pennsylvania, USA

    Alberto Bressan

  2. , Oxford Centre for Nonlinear PDE, Mathema, University of Oxford, St Giles 24–29, Oxford, OX1 3LB, Montserrat

    Gui-Qiang G. Chen

  3. , School of Mathematics, University of Minnesota, Church Street S.E, 206, Minneapolis, 55455, Minnesota, USA

    Marta Lewicka

  4. , Department of Mathematics, University of Pittsburgh, Pittsburgh, 15260, Pennsylvania, USA

    Dehua Wang

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Lefloch, P.G. (2011). Hyperbolic Conservation Laws on Spacetimes. In: Bressan, A., Chen, GQ., Lewicka, M., Wang, D. (eds) Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications, vol 153. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9554-4_21

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