Structure of entropy solutions for multi--dimensional scalar conservation laws
Camillo De Lellis, Felix Otto,and Michael Westdickenberg
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Submission date: 04. Nov. 2002 (revised version: November 2002)
published in: Archive for rational mechanics and analysis, 170 (2003) 2, p. 137-184
DOI number (of the published article): 10.1007/s00205-003-0270-9
MSC-Numbers: 35D10, 35L65, 35L67, 28A75
Keywords and phrases: conservation laws, entropy solutions, partial regularity, shocks, rectifiability
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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.