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Structure of Entropy Solutions for Multi-Dimensional Scalar Conservation Laws

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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.

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Lellis, C., Otto, F. & Westdickenberg, M. Structure of Entropy Solutions for Multi-Dimensional Scalar Conservation Laws. Arch. Rational Mech. Anal. 170, 137–184 (2003). https://doi.org/10.1007/s00205-003-0270-9

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