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arXiv · 2607.17310

Multidimensional scalar conservation laws with discontinuous flux: well-posedness without non-degeneracy

Darko Mitrovic
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Abstract

We establish existence, uniqueness, and local \(L^1\)-stability for multidimensional scalar conservation laws with discontinuous heterogeneous flux, without imposing a non-degeneracy condition on the physical space--time flux. The discontinuity set may be a locally finite family of \(C^2\) hypersurfaces whose intersections and singular points are contained in a closed set of vanishing \((d-1)\)-dimensional Hausdorff measure. The main obstacle is the possible failure of multidimensional compactness when a nontrivial linear combination of the space--time flux components is constant on an interval of states. We show that, for interface problems, this obstruction can be reduced to the physical normal flux. Its flat intervals are collapsed by a normal-flux quotient that preserves the normal flux and hence the Rankine--Hugoniot relation. On the non-flat region, the time and tangential components are replaced by auxiliary polynomial functions of the normal flux, while the physical normal component is left unchanged. The resulting auxiliary space--time vector satisfies the interval non-degeneracy required by Panov's compactness theory. This yields strong compactness and strong one-sided traces of the quotient variables. The interface admissibility condition is obtained by projecting the vanishing-viscosity germ associated with the two one-sided physical normal fluxes. A localized viscous comparison shows that the quotient traces belong to this maximal \(L^1\)-dissipative germ, and a Young-measure contraction argument recovers strong convergence of the physical states. Curved interfaces are treated by flattening, localization, and finite-propagation patching. The theory is first constructed for \(BV\) initial data and then extended by continuity to \(L^1_{\mathrm{loc}}\cap L^\infty\) data taking values in the invariant interval.

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Darko Mitrovic. 2026-07-19. Multidimensional scalar conservation laws with discontinuous flux: well-posedness without non-degeneracy. https://arxiv.org/abs/2607.17310

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