Singular viscosity-exponent limit for one-dimensional compressible Navier-Stokes equations with far field vacuum

Abstract

We consider the limit $δ\to1^+$ 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 $δ\to1^+$ for strong solutions with far field vacuum. With far field vacuum, the diffusion coefficient $ρ^{δ-1}$ does not converge uniformly to $1$ on $\mathbb R$ as $δ\to1^+$. 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 $ρ_0^{(δ-1)/2}\in L^2(\mathbb R)$. 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 $+\infty$ as $δ\to1^+$.