Supercritical Geometric Optics for multidimensional scalar conservation laws
Abstract
This article studies the propagation, near a smooth non-constant ground state, of monophase oscillatory wave trains with small amplitude and very high frequency for multidimensional nonlinear scalar conservation laws. These oscillations lie beyond the classical regime of uniformly bounded variation. Two scenarios may occur. Under suitable compatibility conditions, the oscillations propagate. Otherwise, they are damped through shock formation. In the baseline supercritical regime, we give an explicit criterion distinguishing these two behaviors and justify the corresponding asymptotic descriptions. At higher frequencies, the propagation mechanism becomes more restrictive and is governed by compatibility conditions involving a Lie algebra. Under a natural local constant-rank assumption, we obtain a necessary and sufficient condition for the existence of complete WKB expansions. We also prove cancellation under first-order incompatibility. Perturbing a non-constant background state limits the propagation of small scales compared with the constant-background case. Our constructions contribute to the understanding of the fractional smoothing effect conjectured by Lions, Perthame and Tadmor.
Related
- Uniform positive scalar curvature from supercritical quadratic decay
- On blow-up for the supercritical defocusing nonlinear scalar wave equation
- Noether symmetries and conservation laws of a class of time-dependent multidimensional nonlinear wave equations
- Scalar approach to ARZ-type systems of conservation laws
- Diffractive Geometric Optics for Bloch Wave Packets
BibTeX
@misc{cheverry2026supercritical,
title = {{Supercritical Geometric Optics for multidimensional scalar conservation laws}},
author = {Christophe Cheverry and Rida Harb and Stéphane Junca},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35034},
eprint = {2610.35034},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05836661v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Supercritical Geometric Optics for multidimensional scalar conservation laws
AU - Christophe Cheverry
AU - Rida Harb
AU - Stéphane Junca
PY - 2026
DA - 2026/10/23
PB - HAL
DO - 10.48550/arxiv.2610.35034
UR - https://hal.science/hal-05836661v1
AB - This article studies the propagation, near a smooth non-constant ground state, of monophase oscillatory wave trains with small amplitude and very high frequency for multidimensional nonlinear scalar conservation laws. These oscillations lie beyond the classical regime of uniformly bounded variation. Two scenarios may occur. Under suitable compatibility conditions, the oscillations propagate. Otherwise, they are damped through shock formation. In the baseline supercritical regime, we give an explicit criterion distinguishing these two behaviors and justify the corresponding asymptotic descriptions. At higher frequencies, the propagation mechanism becomes more restrictive and is governed by compatibility conditions involving a Lie algebra. Under a natural local constant-rank assumption, we obtain a necessary and sufficient condition for the existence of complete WKB expansions. We also prove cancellation under first-order incompatibility. Perturbing a non-constant background state limits the propagation of small scales compared with the constant-background case. Our constructions contribute to the understanding of the fractional smoothing effect conjectured by Lions, Perthame and Tadmor.
ER -