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SBV Regularity Results for Solutions to 1D Conservation Laws

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Hyperbolic Conservation Laws and Related Analysis with Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 49))

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Abstract

A well-posedness theory has been established for entropy solutions to strictly hyperbolic systems of conservation laws, in one space variable, with small total variation. We give in this note an introduction to SBV-regularity results: when the characteristic fields are genuinely nonlinear, the derivative of an entropy solution consists only of the absolutely continuous part and of the jump part, while a fractal behavior (the Cantor part) is ruled out. We first review the scalar uniformly convex case, related to the Hopf-Lax formula. We then turn to the case of systems: one has a decay estimate for both positive and negative waves, obtained considering the interaction-cancellation measures and balance measures for the jump part of the waves. When the Cantor part of the time restriction of the entropy solution does not vanish, either the Glimm functional has a downward jump, or there is a cancellation of waves or this wave balance measure is positive, and this can occur at most at countably many times. We then remove the assumption of genuine nonlinearity. The Cantor part is in general present. There are however interesting nonlinear functions of the entropy solution which still enjoy this regularity.

2010 Mathematics Subject Classification 35-02/35-06, 35L65, 35B65

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References

  1. L. Ambrosio, C. De Lellis, A note on admissible solutions of 1d scalar conservation laws and 2d Hamilton-Jacobi equations. J. Hyperbolic Differ. Equ. 31(4), 813–826 (2004)

    Article  Google Scholar 

  2. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems (Clarendon, Oxford, 2000)

    MATH  Google Scholar 

  3. F. Ancona, A. Marson, Existence theory by front tracking for general nonlinear hyperbolic systems. Arch. Ration. Mech. Anal. 185(2), 287–340 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Ancona, K.T. Nguyen, SBV regularity of L ∞ solutions to genuinely nonlinear temple systems of balance laws (in preparation)

    Google Scholar 

  5. S. Bianchini, SBV regularity of genuinely nonlinear hyperbolic systems of conservation laws in one space dimension. Acta Math. Sci. 32B(1), 380–388 (2012)

    Article  MathSciNet  Google Scholar 

  6. S. Bianchini, L. Caravenna, SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension. Commun. Math. Phys. 313(1), 1–33 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Bianchini, D. Tonon, SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t,x). Commun. Pure Appl. Anal. 10(6), 1549–1566 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Bianchini, D. Tonon, SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian. J. Math. Anal. Appl. 391(1), 190–208 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Bianchini, L. Yu, SBV-like regularity for general hyperbolic systems of conservation laws in one space dimension. arXiv:1202.2680

    Google Scholar 

  10. S. Bianchini, C. De Lellis, R. Robyr, SBV regularity for Hamilton-Jacobi equations in \({\mathbb{R}}^{n}\). Arch. Ration. Mech. Anal. 200(3), 1003–1021 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem (Oxford University Press, Oxford, 2000)

    Google Scholar 

  12. A. Bressan, R. Colombo, Decay of positive waves in nonlinear systems of conservation laws. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26(2), 133–160 (1998)

    MathSciNet  MATH  Google Scholar 

  13. A. Bressan, P.G. LeFloch, Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws. Indiana Univ. Math. J. 48(1), 43–84 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Bressan, T. Yang, A sharp decay estimate for nonlinear positive waves. SIAM J. Math. Anal. 36, 659–677 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. C.M. Dafermos, Hyperbolic Conservation Laws in Continuous Physics (Springer, New York, 2000)

    Book  Google Scholar 

  16. C.M. Dafermos, Wave fans are special. Acta Math. Appl. Sin. English Ser. 24(3), 369–374 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)

    Google Scholar 

  18. J. Glimm, P.D. Lax, Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws. Memoirs of the American Mathematical Society, vol. 101 (American Mathematical Society, Providence, 1970), pp. 95–172

    Google Scholar 

  19. B.H. Li, Global structure of shock waves. Sci. Sin. 22, 979–990 (1979)

    MATH  Google Scholar 

  20. O. Oleinik, Discontinuous solutions of nonlinear differential equations. Am. Math. Soc. Transl. 26, 95–172 (1963). Translation of Uspekhi Mat. Nauk vol. 12, 3(75), 3–73 (1957)

    Google Scholar 

  21. R. Robyr, SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function. J. Hyperbolic Differ. Equ. 5(2), 449–475 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. D. Schaeffer, A regularity theorem for conservation laws. Adv. Math. 11, 368–386 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author wishes to thank Gui-Qiang G. Chen and the anonymous referee for interesting comments. She has been supported by the UK EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1).

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX1 2LB, UK

    Laura Caravenna

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  1. Laura Caravenna
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Correspondence to Laura Caravenna .

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Editors and Affiliations

  1. University of Oxford Mathematical Institute, Oxford, United Kingdom

    Gui-Qiang G. Chen

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Helge Holden

  3. University of Oslo Centre of Mathematics for Applications, Oslo, Norway

    Kenneth H. Karlsen

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Caravenna, L. (2014). SBV Regularity Results for Solutions to 1D Conservation Laws. In: Chen, GQ., Holden, H., Karlsen, K. (eds) Hyperbolic Conservation Laws and Related Analysis with Applications. Springer Proceedings in Mathematics & Statistics, vol 49. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39007-4_3

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