Calculus of Variations and Geometric Measure Theory

C. De Lellis - F. Otto - M. Westdickenberg

Structure of entropy solutions for multi--dimensional conservation laws

created on 14 Nov 2002
modified by delellis on 03 May 2011


Published Paper

Inserted: 14 nov 2002
Last Updated: 3 may 2011

Journal: Archive for Rational Mechanics and Analysis
Volume: 170
Pages: 137-184
Year: 2003


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.

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Keywords: Partial regularity, Rectifiability, conservation laws, entropy solutions, shocks