Time-asymptotic stability of generic Riemann solutions for viscous-dispersive heat-conducting compressible fluids
Abstract
We establish the time-asymptotic stability of generic Riemann solutions for the one-dimensional Navier-Stokes-Fourier-Korteweg (NSFK) system, which describes the dynamics of viscous, heat-conducting, compressible fluids with internal capillarity. We prove that, for the adiabatic exponent γ∈(1,2], the solution to the NSFK system converges uniformly in space, as time tends to infinity, to a composite wave consisting of a rarefaction wave, a viscous contact wave, and a viscous-dispersive shock with a time-dependent shift, provided that the initial perturbation and the strengths of the contact wave and the shock are sufficiently small. In particular, we do not impose any smallness condition on the strength of the rarefaction wave. Because of the capillarity, the shock is viscous-dispersive and its profile equation is of third order. We treat the shock by the method of a-contraction with shifts introduced by Kang and Vasseur. This is the first result on the time-asymptotic stability of a viscous-dispersive shock, and of a generic Riemann solution, for the NSFK system.
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BibTeX
@misc{han2026timeasymptotic,
title = {{Time-asymptotic stability of generic Riemann solutions for viscous-dispersive heat-conducting compressible fluids}},
author = {Sungho Han and Jeongho Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29859},
eprint = {2610.29859},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29859},
}
RIS
TY - UNPB
TI - Time-asymptotic stability of generic Riemann solutions for viscous-dispersive heat-conducting compressible fluids
AU - Sungho Han
AU - Jeongho Kim
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29859
UR - https://arxiv.org/abs/2610.29859
AB - We establish the time-asymptotic stability of generic Riemann solutions for the one-dimensional Navier-Stokes-Fourier-Korteweg (NSFK) system, which describes the dynamics of viscous, heat-conducting, compressible fluids with internal capillarity. We prove that, for the adiabatic exponent γ∈(1,2], the solution to the NSFK system converges uniformly in space, as time tends to infinity, to a composite wave consisting of a rarefaction wave, a viscous contact wave, and a viscous-dispersive shock with a time-dependent shift, provided that the initial perturbation and the strengths of the contact wave and the shock are sufficiently small. In particular, we do not impose any smallness condition on the strength of the rarefaction wave. Because of the capillarity, the shock is viscous-dispersive and its profile equation is of third order. We treat the shock by the method of a-contraction with shifts introduced by Kang and Vasseur. This is the first result on the time-asymptotic stability of a viscous-dispersive shock, and of a generic Riemann solution, for the NSFK system.
ER -