Well-posedness of the 3-D compressible Navier-Stokes equations with density-dependent viscosities in exterior domains with far-field vacuum

Hairong Liu; Hua Zhong

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arXiv · 2512.07614

Well-posedness of the 3-D compressible Navier-Stokes equations with density-dependent viscosities in exterior domains with far-field vacuum

Hairong Liu·Hua Zhong

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Abstract

This paper investigates the local existence and uniqueness of strong solutions to the three-dimensional compressible Navier-Stokes equations with density-dependent viscosities in exterior domains. When both the shear and bulk viscosity coefficients depend on the density in a power law ($\rho^{\delta}$ with $0<\delta<1$) and Navier-slip boundary condition on the velocity is imposed, base on a reformulation of the problem using new variables to handle the degeneracy near vacuum, we establish the local well-posedness of regular solutions with far-field vacuum in inhomogeneous Sobolev spaces. Compared to the Cauchy problem, the initial-boundary value problem requires establishing non-standard weighted estimates and handling the unavailability of boundary conditions for higher-order terms. Our approach addresses these challenges via the conormal space technique.

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Hairong Liu, Hua Zhong. 2025-12-08. Well-posedness of the 3-D compressible Navier-Stokes equations with density-dependent viscosities in exterior domains with far-field vacuum. https://arxiv.org/abs/2512.07614

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