Uniform Exponential Stability for Finite Energy Weak Solutions to the Barotropic Compressible Navier--Stokes Equations with Large Potential External Forces
Abstract
We study the large-time behavior of Lions--Feireisl finite energy weak solutions to the barotropic compressible Navier--Stokes equations in three-dimensional bounded domains with Dirichlet boundary conditions and large potential external forces. For γ>3/2, we prove that these weak solutions converge exponentially to the steady state as time tends to infinity, provided that the steady density is bounded away from zero. The main novelty of our analysis is a compactness argument based on Young measures. Furthermore, we prove that the Young measures generated by the density are Dirac measures. Together with the extra integrability of the density, this yields strong convergence to the nonconstant steady density and uniform smallness of the energy w for sufficiently large time. We use Taylor expansion and the Bogovskii operator to construct a Lyapunov functional, and use this smallness of w to obtain exponential decay with a time T0, which is independent of the particular weak solution, thereby answering the question raised by Kang, Peng and Xiong in \cite[Remark~1.1]{KPX}. Corresponding decay estimates for the density, momentum, kinetic energy and the viscous dissipation integrated from large times to infinity also hold, allowing vacuum.
Related
- On the stability of the solutions to compressible Navier-Stokes equations
- On the intrinsic hydrostatic energy structure of the barotropic compressible primitive equations
- Uniqueness of solutions to the 2D compressible Navier-Stokes equations for bounded densities
- Shock solutions of a 3D spherical piston problem for the compressible Euler equations
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BibTeX
@misc{song2026uniform,
title = {{Uniform Exponential Stability for Finite Energy Weak Solutions to the Barotropic Compressible Navier--Stokes Equations with Large Potential External Forces}},
author = {Xin Song and Chengfeng Xiong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35320},
eprint = {2610.35320},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35320},
}
RIS
TY - UNPB
TI - Uniform Exponential Stability for Finite Energy Weak Solutions to the Barotropic Compressible Navier--Stokes Equations with Large Potential External Forces
AU - Xin Song
AU - Chengfeng Xiong
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35320
UR - https://arxiv.org/abs/2610.35320
AB - We study the large-time behavior of Lions--Feireisl finite energy weak solutions to the barotropic compressible Navier--Stokes equations in three-dimensional bounded domains with Dirichlet boundary conditions and large potential external forces. For γ>3/2, we prove that these weak solutions converge exponentially to the steady state as time tends to infinity, provided that the steady density is bounded away from zero. The main novelty of our analysis is a compactness argument based on Young measures. Furthermore, we prove that the Young measures generated by the density are Dirac measures. Together with the extra integrability of the density, this yields strong convergence to the nonconstant steady density and uniform smallness of the energy w for sufficiently large time. We use Taylor expansion and the Bogovskii operator to construct a Lyapunov functional, and use this smallness of w to obtain exponential decay with a time T_0, which is independent of the particular weak solution, thereby answering the question raised by Kang, Peng and Xiong in \cite[Remark~1.1]{KPX}. Corresponding decay estimates for the density, momentum, kinetic energy and the viscous dissipation integrated from large times to infinity also hold, allowing vacuum.
ER -