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Existence and uniqueness of viscous-dispersive shock profiles for the Navier-Stokes-Fourier-Korteweg system
Abstract
We establish the existence and uniqueness, up to translation, of monotone viscous-dispersive shock profiles for the one-dimensional Navier-Stokes-Fourier-Korteweg system, provided the two end states satisfy the Rankine-Hugoniot and Lax entropy conditions and the shock strength is small. Taking the shock strength as a singular perturbation parameter, we reduce the profile equations by Fenichel's geometric singular perturbation theory to a scalar equation on a slow manifold and obtain the profile as a heteroclinic orbit. We also give pointwise estimates of the profiles and their derivatives up to fourth order, with explicit leading-order terms for the first derivatives.
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BibTeX
@misc{han2026existence,
title = {{Existence and uniqueness of viscous-dispersive shock profiles for the Navier-Stokes-Fourier-Korteweg system}},
author = {Sungho Han},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16874},
eprint = {2610.16874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16874},
}
RIS
TY - UNPB
TI - Existence and uniqueness of viscous-dispersive shock profiles for the Navier-Stokes-Fourier-Korteweg system
AU - Sungho Han
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.16874
UR - https://arxiv.org/abs/2610.16874
AB - We establish the existence and uniqueness, up to translation, of monotone viscous-dispersive shock profiles for the one-dimensional Navier-Stokes-Fourier-Korteweg system, provided the two end states satisfy the Rankine-Hugoniot and Lax entropy conditions and the shock strength is small. Taking the shock strength as a singular perturbation parameter, we reduce the profile equations by Fenichel's geometric singular perturbation theory to a scalar equation on a slow manifold and obtain the profile as a heteroclinic orbit. We also give pointwise estimates of the profiles and their derivatives up to fourth order, with explicit leading-order terms for the first derivatives.
ER -