Singular viscosity-exponent limit for one-dimensional compressible Navier-Stokes equations with far field vacuum
Abstract
We consider the limit δ→1+ for the one-dimensional compressible Navier-Stokes equations with viscosity μ(ρ)=ρδ and far field vacuum. To the best of our knowledge, this is the first result on the viscosity-exponent limit δ→1+ for strong solutions with far field vacuum. With far field vacuum, the diffusion coefficient ρδ−1 does not converge uniformly to $1$ on ℝ as δ→1+. The main difficulty is to obtain uniform estimates for this limit. By introducing some new weighted density variables, we establish local well-posedness and uniform estimates without requiring ρ0(δ−1)/2∈ L2(ℝ). We prove strong convergence to the known global δ=1 solution obtained in \cite{WenZhangDCDS2025} on a local time interval independent of δ. With one additional order of regularity of the initial data and the limit solution, the convergence holds on every fixed finite time interval, and the maximal life-spans of the δ>1 solutions tend to +∞ as δ→1+.
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BibTeX
@misc{zhang2026singular,
title = {{Singular viscosity-exponent limit for one-dimensional compressible Navier-Stokes equations with far field vacuum}},
author = {Xingyang Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07790},
eprint = {2611.07790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07790},
}
RIS
TY - UNPB
TI - Singular viscosity-exponent limit for one-dimensional compressible Navier-Stokes equations with far field vacuum
AU - Xingyang Zhang
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07790
UR - https://arxiv.org/abs/2611.07790
AB - We consider the limit δ→1^+ for the one-dimensional compressible Navier-Stokes equations with viscosity μ(ρ)=ρ^δ and far field vacuum. To the best of our knowledge, this is the first result on the viscosity-exponent limit δ→1^+ for strong solutions with far field vacuum. With far field vacuum, the diffusion coefficient ρ^δ−1 does not converge uniformly to $1$ on ℝ as δ→1^+. The main difficulty is to obtain uniform estimates for this limit. By introducing some new weighted density variables, we establish local well-posedness and uniform estimates without requiring ρ_0^(δ−1)/2∈ L^2(ℝ). We prove strong convergence to the known global δ=1 solution obtained in \cite{WenZhangDCDS2025} on a local time interval independent of δ. With one additional order of regularity of the initial data and the limit solution, the convergence holds on every fixed finite time interval, and the maximal life-spans of the δ>1 solutions tend to +∞ as δ→1^+.
ER -