We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
95/2002
Structure of entropy solutions for multi--dimensional scalar conservation laws
Camillo De Lellis, Felix Otto and Michael Westdickenberg
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.
