Archive for Rational Mechanics and Analysis

, Volume 170, Issue 2, pp 137–184

Structure of Entropy Solutions for Multi-Dimensional Scalar Conservation Laws

  • Camillo De Lellis
  • Felix Otto
  • Michael Westdickenberg

DOI: 10.1007/s00205-003-0270-9

Cite this article as:
Lellis, C., Otto, F. & Westdickenberg, M. Arch. Rational Mech. Anal. (2003) 170: 137. doi:10.1007/s00205-003-0270-9


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.

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