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MiS Preprint

Structure of entropy solutions for multi--dimensional scalar conservation laws

Camillo De Lellis, Felix Otto and Michael Westdickenberg


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.

MSC Codes:
35D10, 35L65, 35L67, 28A75
conservation laws, entropy solutions, partial regularity, shocks, rectifiability

Related publications

2003 Repository Open Access
Camillo De Lellis, Felix Otto and Michael Westdickenberg

Structure of entropy solutions for multi-dimensional scalar conservation laws

In: Archive for rational mechanics and analysis, 170 (2003) 2, pp. 137-184