Uniform Stability of degenerate Oleinik shock and rarefaction waves under large perturbations
Abstract
We study the uniform stability of composite waves consisting of a degenerate Oleinik shock and a rarefaction wave for scalar viscous conservation laws. We consider cubic-type fluxes and arbitrarily large L2 initial perturbations of the composite wave within the H1 strong-solution class. We obtain a uniform L2 estimate for such large perturbations. In particular, the estimate implies the contraction of a degenerate Oleinik shock, up to a dynamical shift. As a further consequence, the uniformity with respect to viscosity implies that the associated self-similar solution, composed of the degenerate shock and rarefaction, is stable in the class of inviscid limits. The main analytical difficulty is that the standard weighted Poincaré inequality underlying the a-contraction method fails to yield the required coercivity at the linear level. We overcome this obstruction by establishing a new degenerate Hardy--Poincaré inequality adapted to the degenerate structure of the shock profile. Combined with an exact nonlinear decomposition, this linear inequality yields a strict coercivity estimate that controls the quadratic energy together with the higher-order nonlinear terms, without any smallness assumption on the perturbation.
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BibTeX
@misc{kang2026uniform,
title = {{Uniform Stability of degenerate Oleinik shock and rarefaction waves under large perturbations}},
author = {Moon-Jin Kang and Bongsuk Kwon and Wanyong Shim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26351},
eprint = {2609.26351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26351},
}
RIS
TY - UNPB
TI - Uniform Stability of degenerate Oleinik shock and rarefaction waves under large perturbations
AU - Moon-Jin Kang
AU - Bongsuk Kwon
AU - Wanyong Shim
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26351
UR - https://arxiv.org/abs/2609.26351
AB - We study the uniform stability of composite waves consisting of a degenerate Oleinik shock and a rarefaction wave for scalar viscous conservation laws. We consider cubic-type fluxes and arbitrarily large L^2 initial perturbations of the composite wave within the H^1 strong-solution class. We obtain a uniform L^2 estimate for such large perturbations. In particular, the estimate implies the contraction of a degenerate Oleinik shock, up to a dynamical shift. As a further consequence, the uniformity with respect to viscosity implies that the associated self-similar solution, composed of the degenerate shock and rarefaction, is stable in the class of inviscid limits. The main analytical difficulty is that the standard weighted Poincaré inequality underlying the a-contraction method fails to yield the required coercivity at the linear level. We overcome this obstruction by establishing a new degenerate Hardy--Poincaré inequality adapted to the degenerate structure of the shock profile. Combined with an exact nonlinear decomposition, this linear inequality yields a strict coercivity estimate that controls the quadratic energy together with the higher-order nonlinear terms, without any smallness assumption on the perturbation.
ER -