Asymptotic stability of viscous shock for the outflow problem of 3D Navier-Stokes system
Abstract
We establish the asymptotic stability of planar viscous shock waves for the outflow problem of the three-dimensional barotropic compressible Navier--Stokes equations in a half-space, with periodic boundary conditions imposed in the transverse directions. For a weak shock located sufficiently far from the boundary, we prove that, under small perturbations in H2, the outflow problem admits a unique global-in-time solution and that the solution converges uniformly to the viscous shock, up to a dynamical shift, whose velocity time-asymptotically decays. This provides the first shock-stability result for the multidimensional outflow problem. Our proof combines the a-contraction method and higher-order energy estimates adapted to the half-space boundary. The boundary trace remaining in the zeroth-order estimate is controlled by differentiating the perturbation system in the tangential directions and exploiting the favorable sign of the outflow flux. The highest-order normal derivatives are then recovered from the time and tangential derivatives through the momentum equation, while the boundary terms generated by the dynamical shift are controlled using the exponential decay of the viscous shock tail.
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BibTeX
@misc{kang2026asymptotic,
title = {{Asymptotic stability of viscous shock for the outflow problem of 3D Navier-Stokes system}},
author = {Moon-Jin Kang and Hyeonseop Kim and Hobin Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07071},
eprint = {2609.07071},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07071},
}
RIS
TY - UNPB
TI - Asymptotic stability of viscous shock for the outflow problem of 3D Navier-Stokes system
AU - Moon-Jin Kang
AU - Hyeonseop Kim
AU - Hobin Lee
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07071
UR - https://arxiv.org/abs/2609.07071
AB - We establish the asymptotic stability of planar viscous shock waves for the outflow problem of the three-dimensional barotropic compressible Navier--Stokes equations in a half-space, with periodic boundary conditions imposed in the transverse directions. For a weak shock located sufficiently far from the boundary, we prove that, under small perturbations in H^2, the outflow problem admits a unique global-in-time solution and that the solution converges uniformly to the viscous shock, up to a dynamical shift, whose velocity time-asymptotically decays. This provides the first shock-stability result for the multidimensional outflow problem. Our proof combines the a-contraction method and higher-order energy estimates adapted to the half-space boundary. The boundary trace remaining in the zeroth-order estimate is controlled by differentiating the perturbation system in the tangential directions and exploiting the favorable sign of the outflow flux. The highest-order normal derivatives are then recovered from the time and tangential derivatives through the momentum equation, while the boundary terms generated by the dynamical shift are controlled using the exponential decay of the viscous shock tail.
ER -