Regularizing Effect of Nonlinearity in Multidimensional Scalar Conservation Laws

  • Gianluca Crippa
  • Felix Otto
  • Michael Westdickenberg
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 5)


Weak Solution Bounded Variation Entropy Solution Entropy Condition Kinetic Formulation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Ambrosio, C. De Lellis, and J. Maly. On the chain rule for the divergence of BV like vector fields: Applications, partial results, open problems. In Perspectives in Nonlinear Partial Differential Equations: in honor of Haim Brezis. Birkhäuser, 2006.Google Scholar
  2. 2.
    L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000.zbMATHGoogle Scholar
  3. 3.
    M. Bézard. Régularité L p précisée des moyennes dans les équations de transport. Bull. Soc. Math. France, 122(1):29–76, 1994.zbMATHMathSciNetGoogle Scholar
  4. 4.
    F. Bouchut. Hypoelliptic regularity in kinetic equations. J. Math. Pures Appl. (9), 81(11):1135–1159, 2002.zbMATHMathSciNetGoogle Scholar
  5. 5.
    F. Bouchut and F. James. Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Comm. Partial Differential Equations, 24(11–12):2173–2189, 1999.zbMATHMathSciNetGoogle Scholar
  6. 6.
    Y. Brenier. Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal., 21(6):1013–1037, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    G.-Q. Chen and H. Frid. Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal., 147(2):89–118, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G.-Q. Chen and M. Rascle. Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws. Arch. Ration. Mech. Anal., 153(3):205–220, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    K. S. Cheng. A regularity theorem for a nonconvex scalar conservation law. J. Differential Equations, 61(1):79–127, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C. M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2000.zbMATHGoogle Scholar
  11. 11.
    C. De Lellis, F. Otto, and M. Westdickenberg. Structure of entropy solutions for multi-dimensional scalar conservation laws. Arch. Ration. Mech. Anal., 170(2):137–184, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    C. De Lellis and T. Rivière. The rectifiability of entropy measures in one space dimension. J. Math. Pures Appl. (9), 82(10):1343–1367, 2003.zbMATHMathSciNetGoogle Scholar
  13. 13.
    R. J. DiPerna and P.-L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. (2), 130(2):321–366, 1989.CrossRefMathSciNetGoogle Scholar
  14. 14.
    R. J. DiPerna, P.-L. Lions, and Y. Meyer. L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire, 8(3–4):271–287, 1991.zbMATHMathSciNetGoogle Scholar
  15. 15.
    W. Gautschi. On inverses of Vandermonde and confluent Vandermonde matrices. Numer. Math., 4:117–123, 1962.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    F. Golse, P.-L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 76(1):110–125, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    D. Hoff. The sharp form of Oleĭnik’s entropy condition in several space variables. Trans. Amer. Math. Soc., 276(2):707–714, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    P.-E. Jabin and B. Perthame. Regularity in kinetic formulations via averaging lemmas. ESAIM Control Optim. Calc. Var., 8:761–774 (electronic), 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    S. N. Kružkov. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.), 81 (123):228–255, 1970.MathSciNetGoogle Scholar
  20. 20.
    P.-L. Lions, B. Perthame, and E. Tadmor. A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc., 7(1):169–191, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    P. Mattila. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995.zbMATHGoogle Scholar
  22. 22.
    O. A. Oleĭnik. Discontinuous solutions of non-linear differential equations. Uspehi Mat. Nauk (N.S.), 12(3(75)):3–73, 1957.Google Scholar
  23. 23.
    E. Yu. Panov. Existence of strong traces for generalized solutions of multidimensional scalar conservation laws. J. Hyperbolic Differ. Equ., 2(4):885–908, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    E. Tadmor and T. Tao. Velocity averaging, kinetic formulations, and regularizing effects in quasilinear PDEs. Comm. Pure Appl. Math., 2006.Google Scholar
  25. 25.
    A. Vasseur. Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal., 160(3):181–193, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    M. Westdickenberg. Some new velocity averaging results. SIAM J. Math. Anal., 33(5):1007–1032 (electronic), 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    K. Zumbrun. Decay rates for nonconvex systems of conservation laws. Comm. Pure Appl. Math., 46(3):353–386, 1993.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Gianluca Crippa
    • Felix Otto
      • Michael Westdickenberg

        There are no affiliations available

        Personalised recommendations