Local well-posedness of strong solutions to the compressible Navier-Stokes equations with degenerate viscosities and far field vacuum in 3D exterior domains
Jiaxu Li, Boqiang Lü, Bing Yuan
TL;DR
The paper proves local existence and uniqueness of strong solutions to the 3D isentropic compressible Navier–Stokes equations with density-dependent, degenerate viscosities in an exterior domain and far-field vacuum under Navier-slip boundary conditions. A reformulation dividing by $\rho^{\delta}$ leads to a strongly parabolic structure that enables uniform a priori estimates on annular domains and a limiting passage to the exterior domain. The authors establish a blow-up criterion, showing that global existence hinges on controlling key norms such as $\|\mathcal{D}(u)\|_{L^1(0,T;L^\infty)}$, $\|\rho^{(\delta-1)/2}\|_{L^\infty(0,T;L^6(\Omega_0))}$, and $\|\nabla\rho^{(\delta-1)/2}\|_{L^\infty(0,T;D^{1,2})}$. A critical feature is that the admissible range of $\delta$ is independent of the gas constant $\gamma$, and a positive lower bound for density near the boundary is obtained to manage boundary terms. The results extend local well-posedness for degenerate-viscosity flows from Cauchy problems to 3D exterior-domain IBVPs and provide a framework for studying boundary effects with vacuum.
Abstract
The isentropic compressible Navier-Stokes system subject to the Navier-slip boundary conditions is considered in a general three-dimensional exterior domain. For the density approaches far-field vacuum initially and the viscosities are power functions of the density(ρ^δ with 0 < δ< 1), the local well-posedness of strong solutions is established in this paper. In particular, the method we adopt can not only simultaneously handle the difficulties caused by boundary terms and far-field vacuum, but also make the selection of δ independent of the gas coefficient γ.