Preprint

Inserted: 4 oct 2026
Last Updated: 4 oct 2026

Year: 2026

Abstract:

We construct a bounded weak solution of a one-dimensional scalar conservation law, with a $C^2$ flux that has non-zero second derivative at some state, whose non-Lebesgue points outside the entropy jump set form a compact, purely $1$-unrectifiable set $E$ of Hausdorff dimension $1$ that is not $\mathcal H^1$-$\sigma$-finite. For each convex entropy, the entropy production is a finite signed measure, and the one-dimensional density of its total variation vanishes uniformly on $E$. Moreover, the mean oscillation of the solution vanishes uniformly on $E$, and $E$ is the image of a product of two Cantor sets under a bi-Lipschitz map. In particular, we answer a question of Marconi (Calc. Var. Partial Differential Equations, 2022) about Burgers' equation, and disprove a conjecture of De Lellis and Otto (J. Eur. Math. Soc., 2003) about the eikonal equation.

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