Year: 2026
Type: article
Abstract: We study the optimal regularizing effect for scalar balance laws with strictly convex fluxes and time-dependent nonlinear source terms. In a recent work, we proved that a suitable transversality condition between the source term and the degeneracy set of the flux restores full $BV_{\mathrm{loc}}$ regularity. In this note, we show that this condition is optimal. If the flux has polynomial degeneracy and the source vanishes along a degenerate state on a nontrivial time interval, then the full $BV_{\mathrm{loc}}$ regularizing effect cannot be expected in general. We construct an entropy solution showing that the fractional threshold $BV_{\mathrm{loc}}^{1/p}$, already known from the regularizing theory, cannot be improved. The same construction also yields the corresponding sharp obstruction in the Sobolev scale.
Author: Rida Harb
Institutions: Université Côte d'Azur, Laboratoire Jean-Alexandre Dieudonné
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W7166621823