Archive for Rational Mechanics and Analysis

, Volume 170, Issue 2, pp 137–184 | Cite as

Structure of Entropy Solutions for Multi-Dimensional Scalar Conservation Laws

  • Camillo De Lellis
  • Felix Otto
  • Michael Westdickenberg
Article

Abstract.

An entropy solution u of a multi-dimensional scalar conservation law is not necessarily in BV, even if the conservation law is genuinely nonlinear. We show that u nevertheless has the structure of a BV function in the sense that the shock location is codimension-one rectifiable. This result highlights the regularizing effect of genuine nonlinearity in a qualitative way; it is based on the locally finite rate of entropy dissipation. The proof relies on the geometric classification of blow-ups in the framework of the kinetic formulation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000Google Scholar
  2. 2.
    Ambrosio, L., Kirchheim, B., Lecumberry, M., Rivière, T.: The local structure of Néel Walls. Nonlinear Problems in Mathematical Physics and Related Topics II. In Honour of Professor O.A. Ladyzhenskaya. International Mathematical Series vol. 2. Kluwer Academic, 2002Google Scholar
  3. 3.
    Ambrosio, L., Lecumberry, M., Rivière, T.: Viscosity property of minimizing micromagnetic configurations. Comm. Pure Appl. Math. 56, 681–688 (2003)MATHGoogle Scholar
  4. 4.
    Bénilan, P., Crandall, M.: Regularizing effect of homogeneous evolution equations. Contributions to analysis and geometry (Baltimore, Md., 1980), Johns Hopkins Univ. Press, 1981, pp. 23–39Google Scholar
  5. 5.
    Brenner, P.: The Cauchy problem for symmetric hyperbolic systems in L p. Math. Scand. 19, 27–37 (1966)MATHGoogle Scholar
  6. 6.
    Bressan, A.: Hyperbolic systems of conservation laws. The one dimensional Cauchy problem. Oxford Lecture Series in Mathematics and its Applications vol. 20. Oxford University Press, Oxford, 2000Google Scholar
  7. 7.
    Chen, G.-Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Rational Mech. Anal. 147, 89–118 (1999)MathSciNetMATHGoogle Scholar
  8. 8.
    Chen, G.-Q., Rascle, M.: Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws. Arch. Rational Mech. Anal. 153, 205–220 (2000)MathSciNetMATHGoogle Scholar
  9. 9.
    Cheverry, C.: Regularizing effects for multidimensional scalar conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 17, 413–472 (2000)MathSciNetMATHGoogle Scholar
  10. 10.
    Dafermos, C.: Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenschaften vol. 325. Springer-Verlag, Berlin, 2000Google Scholar
  11. 11.
    De Lellis, C., Otto, F.: Structure of entropy solutions to the eikonal equation. J. Eur. Math. Soc. 5, 107–145 (2003)MATHGoogle Scholar
  12. 12.
    Di Perna, R., Lions, P.-L., Meyer, Y.: L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 271–287 (1991)MATHGoogle Scholar
  13. 13.
    Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992Google Scholar
  14. 14.
    Golse, S., Lions, P.-L., Perthame, B., Sentis, R.: Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76, 110–125 (1988)MathSciNetMATHGoogle Scholar
  15. 15.
    Jabin, P.-E., Otto, F., Perthame, B.: Line-energy Ginzburg-Landau models: zero-energy states. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 1, 187–202 (2002)Google Scholar
  16. 16.
    Jabin, P.-E., Perthame, B.: Regularity in kinetic formulations via averaging lemmas: A tribute to J.-L. Lions. ESAIM Control Optim. Calc. Var. 8, 761–774 (2002)Google Scholar
  17. 17.
    Kruzkov, S.N.: First order quasilinear equations in several independent variables. Math USSR–Sbornik 10, 217–243 (1970)Google Scholar
  18. 18.
    Lecumberry, M., Rivière, T.: The rectifiability of shock waves for the solutions of genuinely non-linear scalar conservation laws in 1+1 D. Preprint 44 in http://www.math.ethz.ch/riviere/pub.html. Google Scholar
  19. 19.
    Lions, P.-L., Perthame, B., Tadmor, E.: A kinetic formulation of multidimensional scalar conservation laws and related questions. J. AMS 7, 169–191 (1994)MathSciNetMATHGoogle Scholar
  20. 20.
    Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics vol. 44. Cambridge University Press, Cambridge, 1995Google Scholar
  21. 21.
    Oleinik, O.A.: Discontinuous solutions of nonlinear differential equations. Transl. AMS Ser. 2 26, 95–172 (1957)Google Scholar
  22. 22.
    Otto, F.: A regularizing effect of nonlinear transport equations. Quart. Appl. Math. 56, 355–375 (1998)MathSciNetMATHGoogle Scholar
  23. 23.
    Rudin, W.: Functional Analysis. International Series in Pure and Applied Math., McGraw-Hill, second edition, 1991Google Scholar
  24. 24.
    Serre, D.: Systems of conservation laws. 1. Hyperbolicity, entropies, shock waves. Cambridge University Press, Cambridge, 2000Google Scholar
  25. 25.
    Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Herriot-Watt Symposium, Vol~IV, Res. Notes in Math. 39, 1979, pp. 136–212Google Scholar
  26. 26.
    Vasseur, A.: Strong traces for solutions of multidimensional scalar conservation laws. Arch. Rational Mech. Anal. 160, 181–193 (2001)MathSciNetMATHGoogle Scholar
  27. 27.
    Westdickenberg, M.: Some new velocity averaging results. SIAM J. Math. Anal. 33, 1007–1032 (2002)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Camillo De Lellis
    • 1
  • Felix Otto
    • 2
  • Michael Westdickenberg
    • 3
  1. 1.Max-Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Institute for Applied MathematicsUniversity of BonnBonnGermany
  3. 3.Institute for Applied MathematicsUniversity of BonnBonnGermany

Personalised recommendations