On the stability of the solutions to compressible Navier-Stokes equations
Abstract
This paper investigates the continuous dependence (stability) of solutions to the barotropic compressible Navier--Stokes equations in critical Besov spaces. Using the Lagrangian approach developed in \cite{Danchin2014}, it was shown that, for 1<p<2d, the flow map (a0,u0)↦ (a̅,u̅)=(a∘ X,u∘ X) is Lipschitz continuous from Ḃp,1d/p× Ḃp,1d/p−1 into 𝒞([0,T];Ḃp,1d/p)× Ep(T). However, this result does not directly imply the corresponding continuous dependence in the original Eulerian coordinates because of the low regularity of the critical initial data. Previously, only for p<d, continuous dependence was known only in certain lower-regularity spaces with a loss of one derivative relative to the natural solution spaces arising essentially as a by-product of the uniqueness argument. We close this gap and prove that for 1<p<2d the flow map (a0,u0)↦(a,u) is continuous (not Lipschitz continuous) from Ḃp,1d/p×Ḃp,1d/p−1 to 𝒞([0,T];Ḃp,1d/p)× Ep(T) in Eulerian coordinates without any loss of regularity, which together with the known existence and uniqueness theory \cite{Danchin2014} completes Hadamard well-posedness in the critical spaces.
Related
- Uniform Exponential Stability for Finite Energy Weak Solutions to the Barotropic Compressible Navier--Stokes Equations with Large Potential External Forces
- Uniqueness of solutions to the 2D compressible Navier-Stokes equations for bounded densities
- Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions
- Local and Global Existence of Strong Solutions to the One-Dimensional Compressible Navier--Stokes Equations with General Pressure Law
- Asymptotic behavior of generic Riemann solutions to one-dimensional full compressible Navier-Stokes-Poisson equations
BibTeX
@misc{dong2026stability,
title = {{On the stability of the solutions to compressible Navier-Stokes equations}},
author = {Qiaojie Dong and Min Li and Yatao Li and Minghua Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22749},
eprint = {2609.22749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22749},
}
RIS
TY - UNPB
TI - On the stability of the solutions to compressible Navier-Stokes equations
AU - Qiaojie Dong
AU - Min Li
AU - Yatao Li
AU - Minghua Yang
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22749
UR - https://arxiv.org/abs/2609.22749
AB - This paper investigates the continuous dependence (stability) of solutions to the barotropic compressible Navier--Stokes equations in critical Besov spaces. Using the Lagrangian approach developed in \cite{Danchin2014}, it was shown that, for 1<p<2d, the flow map (a_0,u_0)↦ (a̅,u̅)=(a∘ X,u∘ X) is Lipschitz continuous from Ḃ_p,1^d/p× Ḃ_p,1^d/p−1 into 𝒞([0,T];Ḃ_p,1^d/p)× E_p(T). However, this result does not directly imply the corresponding continuous dependence in the original Eulerian coordinates because of the low regularity of the critical initial data. Previously, only for p<d, continuous dependence was known only in certain lower-regularity spaces with a loss of one derivative relative to the natural solution spaces arising essentially as a by-product of the uniqueness argument. We close this gap and prove that for 1<p<2d the flow map (a_0,u_0)↦(a,u) is continuous (not Lipschitz continuous) from Ḃ_p,1^d/p×Ḃ_p,1^d/p−1 to 𝒞([0,T];Ḃ_p,1^d/p)× E_p(T) in Eulerian coordinates without any loss of regularity, which together with the known existence and uniqueness theory \cite{Danchin2014} completes Hadamard well-posedness in the critical spaces.
ER -